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Properties

Label 2-425-17.15-c1-0-10
Degree 22
Conductor 425425
Sign 0.992−0.125i0.992 - 0.125i
Analytic cond. 3.393643.39364
Root an. cond. 1.842181.84218
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.176 + 0.176i)2-s + (0.629 + 1.51i)3-s − 1.93i·4-s + (−0.156 + 0.377i)6-s + (1.32 + 0.547i)7-s + (0.693 − 0.693i)8-s + (0.210 − 0.210i)9-s + (1.03 − 2.48i)11-s + (2.94 − 1.21i)12-s − 0.174i·13-s + (0.136 + 0.329i)14-s − 3.63·16-s + (3.44 + 2.27i)17-s + 0.0742·18-s + (2.69 + 2.69i)19-s + ⋯
L(s)  = 1  + (0.124 + 0.124i)2-s + (0.363 + 0.876i)3-s − 0.969i·4-s + (−0.0639 + 0.154i)6-s + (0.499 + 0.206i)7-s + (0.245 − 0.245i)8-s + (0.0703 − 0.0703i)9-s + (0.310 − 0.750i)11-s + (0.849 − 0.351i)12-s − 0.0484i·13-s + (0.0364 + 0.0879i)14-s − 0.908·16-s + (0.834 + 0.550i)17-s + 0.0175·18-s + (0.618 + 0.618i)19-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=((0.992−0.125i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=((0.992−0.125i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: 0.992−0.125i0.992 - 0.125i
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ425(151,⋅)\chi_{425} (151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 425, ( :1/2), 0.992−0.125i)(2,\ 425,\ (\ :1/2),\ 0.992 - 0.125i)

Particular Values

L(1)L(1) ≈\approx 1.81878+0.114672i1.81878 + 0.114672i
L(12)L(\frac12) ≈\approx 1.81878+0.114672i1.81878 + 0.114672i
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)
bad5 1 1
17 1+(−3.44−2.27i)T 1 + (-3.44 - 2.27i)T
good2 1+(−0.176−0.176i)T+2iT2 1 + (-0.176 - 0.176i)T + 2iT^{2}
3 1+(−0.629−1.51i)T+(−2.12+2.12i)T2 1 + (-0.629 - 1.51i)T + (-2.12 + 2.12i)T^{2}
7 1+(−1.32−0.547i)T+(4.94+4.94i)T2 1 + (-1.32 - 0.547i)T + (4.94 + 4.94i)T^{2}
11 1+(−1.03+2.48i)T+(−7.77−7.77i)T2 1 + (-1.03 + 2.48i)T + (-7.77 - 7.77i)T^{2}
13 1+0.174iT−13T2 1 + 0.174iT - 13T^{2}
19 1+(−2.69−2.69i)T+19iT2 1 + (-2.69 - 2.69i)T + 19iT^{2}
23 1+(−1.05+2.54i)T+(−16.2−16.2i)T2 1 + (-1.05 + 2.54i)T + (-16.2 - 16.2i)T^{2}
29 1+(5.94−2.46i)T+(20.5−20.5i)T2 1 + (5.94 - 2.46i)T + (20.5 - 20.5i)T^{2}
31 1+(0.188+0.454i)T+(−21.9+21.9i)T2 1 + (0.188 + 0.454i)T + (-21.9 + 21.9i)T^{2}
37 1+(2.19+5.30i)T+(−26.1+26.1i)T2 1 + (2.19 + 5.30i)T + (-26.1 + 26.1i)T^{2}
41 1+(−5.54−2.29i)T+(28.9+28.9i)T2 1 + (-5.54 - 2.29i)T + (28.9 + 28.9i)T^{2}
43 1+(8.00−8.00i)T−43iT2 1 + (8.00 - 8.00i)T - 43iT^{2}
47 1−2.65iT−47T2 1 - 2.65iT - 47T^{2}
53 1+(−8.73−8.73i)T+53iT2 1 + (-8.73 - 8.73i)T + 53iT^{2}
59 1+(−5.72+5.72i)T−59iT2 1 + (-5.72 + 5.72i)T - 59iT^{2}
61 1+(6.41+2.65i)T+(43.1+43.1i)T2 1 + (6.41 + 2.65i)T + (43.1 + 43.1i)T^{2}
67 1+12.3T+67T2 1 + 12.3T + 67T^{2}
71 1+(4.67+11.2i)T+(−50.2+50.2i)T2 1 + (4.67 + 11.2i)T + (-50.2 + 50.2i)T^{2}
73 1+(14.4−5.99i)T+(51.6−51.6i)T2 1 + (14.4 - 5.99i)T + (51.6 - 51.6i)T^{2}
79 1+(−4.94+11.9i)T+(−55.8−55.8i)T2 1 + (-4.94 + 11.9i)T + (-55.8 - 55.8i)T^{2}
83 1+(6.06+6.06i)T+83iT2 1 + (6.06 + 6.06i)T + 83iT^{2}
89 1−11.4iT−89T2 1 - 11.4iT - 89T^{2}
97 1+(−2.12+0.879i)T+(68.5−68.5i)T2 1 + (-2.12 + 0.879i)T + (68.5 - 68.5i)T^{2}
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   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

−10.93657615581148394663115873542, −10.27598680256629495008049361262, −9.445228710601528395572300914253, −8.735106354963821777928523411110, −7.55080499350110094808107154516, −6.21444092878420178142492490636, −5.41479460383212344864650882192, −4.37938545996250424354046141744, −3.29202967985801944558037125588, −1.41865797100854894705232483452, 1.63737588374589903280645595083, 2.86840103022583479557647094527, 4.14630158024493166988754554113, 5.26810366157181470362370925025, 7.04357706677537992517050631747, 7.34213141648999907197423073257, 8.195362966036093791769829298543, 9.169804992868656060339630923095, 10.27807193712079591498015578441, 11.62486775964501421983267319769

Graph of the ZZ-function along the critical line