↑

Properties

Label 2-350-175.108-c1-0-18
Degree 22
Conductor 350350
Sign −0.798−0.602i-0.798 - 0.602i
Analytic cond. 2.794762.79476
Root an. cond. 1.671751.67175
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0523 − 0.998i)2-s + (−1.68 − 1.36i)3-s + (−0.994 + 0.104i)4-s + (2.17 + 0.499i)5-s + (−1.27 + 1.75i)6-s + (−2.26 − 1.36i)7-s + (0.156 + 0.987i)8-s + (0.356 + 1.67i)9-s + (0.384 − 2.20i)10-s + (−4.13 − 0.878i)11-s + (1.82 + 1.18i)12-s + (−1.47 − 2.88i)13-s + (−1.24 + 2.33i)14-s + (−2.99 − 3.82i)15-s + (0.978 − 0.207i)16-s + (−2.51 − 0.963i)17-s + ⋯
L(s)  = 1  + (−0.0370 − 0.706i)2-s + (−0.974 − 0.788i)3-s + (−0.497 + 0.0522i)4-s + (0.974 + 0.223i)5-s + (−0.520 + 0.717i)6-s + (−0.856 − 0.516i)7-s + (0.0553 + 0.349i)8-s + (0.118 + 0.558i)9-s + (0.121 − 0.696i)10-s + (−1.24 − 0.264i)11-s + (0.525 + 0.341i)12-s + (−0.408 − 0.801i)13-s + (−0.332 + 0.623i)14-s + (−0.773 − 0.986i)15-s + (0.244 − 0.0519i)16-s + (−0.608 − 0.233i)17-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=((−0.798−0.602i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+1/2) L(s)=((−0.798−0.602i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: −0.798−0.602i-0.798 - 0.602i
Analytic conductor: 2.794762.79476
Root analytic conductor: 1.671751.67175
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ350(283,⋅)\chi_{350} (283, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 350, ( :1/2), −0.798−0.602i)(2,\ 350,\ (\ :1/2),\ -0.798 - 0.602i)

Particular Values

L(1)L(1) ≈\approx 0.131467+0.392653i0.131467 + 0.392653i
L(12)L(\frac12) ≈\approx 0.131467+0.392653i0.131467 + 0.392653i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.0523+0.998i)T 1 + (0.0523 + 0.998i)T
5 1+(−2.17−0.499i)T 1 + (-2.17 - 0.499i)T
7 1+(2.26+1.36i)T 1 + (2.26 + 1.36i)T
good3 1+(1.68+1.36i)T+(0.623+2.93i)T2 1 + (1.68 + 1.36i)T + (0.623 + 2.93i)T^{2}
11 1+(4.13+0.878i)T+(10.0+4.47i)T2 1 + (4.13 + 0.878i)T + (10.0 + 4.47i)T^{2}
13 1+(1.47+2.88i)T+(−7.64+10.5i)T2 1 + (1.47 + 2.88i)T + (-7.64 + 10.5i)T^{2}
17 1+(2.51+0.963i)T+(12.6+11.3i)T2 1 + (2.51 + 0.963i)T + (12.6 + 11.3i)T^{2}
19 1+(0.765−7.28i)T+(−18.5−3.95i)T2 1 + (0.765 - 7.28i)T + (-18.5 - 3.95i)T^{2}
23 1+(0.420−0.0220i)T+(22.8−2.40i)T2 1 + (0.420 - 0.0220i)T + (22.8 - 2.40i)T^{2}
29 1+(−1.60−2.20i)T+(−8.96+27.5i)T2 1 + (-1.60 - 2.20i)T + (-8.96 + 27.5i)T^{2}
31 1+(2.29+5.15i)T+(−20.7+23.0i)T2 1 + (2.29 + 5.15i)T + (-20.7 + 23.0i)T^{2}
37 1+(−4.14+6.38i)T+(−15.0−33.8i)T2 1 + (-4.14 + 6.38i)T + (-15.0 - 33.8i)T^{2}
41 1+(8.21+2.66i)T+(33.1+24.0i)T2 1 + (8.21 + 2.66i)T + (33.1 + 24.0i)T^{2}
43 1+(5.92+5.92i)T+43iT2 1 + (5.92 + 5.92i)T + 43iT^{2}
47 1+(1.06+2.78i)T+(−34.9+31.4i)T2 1 + (1.06 + 2.78i)T + (-34.9 + 31.4i)T^{2}
53 1+(−3.55+4.38i)T+(−11.0−51.8i)T2 1 + (-3.55 + 4.38i)T + (-11.0 - 51.8i)T^{2}
59 1+(−1.52−1.68i)T+(−6.16+58.6i)T2 1 + (-1.52 - 1.68i)T + (-6.16 + 58.6i)T^{2}
61 1+(6.08+5.47i)T+(6.37+60.6i)T2 1 + (6.08 + 5.47i)T + (6.37 + 60.6i)T^{2}
67 1+(−3.21+8.36i)T+(−49.7−44.8i)T2 1 + (-3.21 + 8.36i)T + (-49.7 - 44.8i)T^{2}
71 1+(−12.4+9.05i)T+(21.9−67.5i)T2 1 + (-12.4 + 9.05i)T + (21.9 - 67.5i)T^{2}
73 1+(12.0−7.84i)T+(29.6−66.6i)T2 1 + (12.0 - 7.84i)T + (29.6 - 66.6i)T^{2}
79 1+(0.551−1.23i)T+(−52.8−58.7i)T2 1 + (0.551 - 1.23i)T + (-52.8 - 58.7i)T^{2}
83 1+(−15.4+2.43i)T+(78.9−25.6i)T2 1 + (-15.4 + 2.43i)T + (78.9 - 25.6i)T^{2}
89 1+(−5.00+5.55i)T+(−9.30−88.5i)T2 1 + (-5.00 + 5.55i)T + (-9.30 - 88.5i)T^{2}
97 1+(13.0+2.06i)T+(92.2+29.9i)T2 1 + (13.0 + 2.06i)T + (92.2 + 29.9i)T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.70738251093723798311173585498, −10.37014522435817171770056613393, −9.451640896712106166341755955675, −8.011214265132695055668088706150, −6.93388055209041532122437442927, −5.93369647876849130780430360775, −5.25641539644828084034460853329, −3.40794731076427289951664718838, −2.05661671127554959427466120125, −0.30391913153978530252926808962, 2.60498423843884149202552915195, 4.66432272552524277775701072504, 5.15166997364201499474761852911, 6.20128290229760344168458840277, 6.87889586994228382913702239953, 8.495708187435949161417744419866, 9.456368476712853812713202327921, 10.03123044774981950447325023482, 10.90014710464042402349650177908, 12.02644500504704636434463460504

Graph of the ZZ-function along the critical line