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Properties

Label 2-4140-15.2-c1-0-12
Degree 22
Conductor 41404140
Sign 0.522−0.852i0.522 - 0.852i
Analytic cond. 33.058033.0580
Root an. cond. 5.749615.74961
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  + (−1.79 + 1.33i)5-s + (−0.390 − 0.390i)7-s + 0.378i·11-s + (−3.06 + 3.06i)13-s + (5.04 − 5.04i)17-s − 1.94i·19-s + (−0.707 − 0.707i)23-s + (1.45 − 4.78i)25-s − 0.264·29-s + 0.305·31-s + (1.22 + 0.182i)35-s + (1.13 + 1.13i)37-s + 4.48i·41-s + (−2.13 + 2.13i)43-s + (5.21 − 5.21i)47-s + ⋯
L(s)  = 1  + (−0.803 + 0.595i)5-s + (−0.147 − 0.147i)7-s + 0.114i·11-s + (−0.848 + 0.848i)13-s + (1.22 − 1.22i)17-s − 0.447i·19-s + (−0.147 − 0.147i)23-s + (0.291 − 0.956i)25-s − 0.0491·29-s + 0.0548·31-s + (0.206 + 0.0307i)35-s + (0.187 + 0.187i)37-s + 0.700i·41-s + (−0.325 + 0.325i)43-s + (0.761 − 0.761i)47-s + ⋯

Functional equation

Λ(s)=(4140s/2 ΓC(s) L(s)=((0.522−0.852i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(4140s/2 ΓC(s+1/2) L(s)=((0.522−0.852i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 41404140    =    22⋅32⋅5⋅232^{2} \cdot 3^{2} \cdot 5 \cdot 23
Sign: 0.522−0.852i0.522 - 0.852i
Analytic conductor: 33.058033.0580
Root analytic conductor: 5.749615.74961
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ4140(737,⋅)\chi_{4140} (737, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4140, ( :1/2), 0.522−0.852i)(2,\ 4140,\ (\ :1/2),\ 0.522 - 0.852i)

Particular Values

L(1)L(1) ≈\approx 1.2610930271.261093027
L(12)L(\frac12) ≈\approx 1.2610930271.261093027
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 1
3 1 1
5 1+(1.79−1.33i)T 1 + (1.79 - 1.33i)T
23 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good7 1+(0.390+0.390i)T+7iT2 1 + (0.390 + 0.390i)T + 7iT^{2}
11 1−0.378iT−11T2 1 - 0.378iT - 11T^{2}
13 1+(3.06−3.06i)T−13iT2 1 + (3.06 - 3.06i)T - 13iT^{2}
17 1+(−5.04+5.04i)T−17iT2 1 + (-5.04 + 5.04i)T - 17iT^{2}
19 1+1.94iT−19T2 1 + 1.94iT - 19T^{2}
29 1+0.264T+29T2 1 + 0.264T + 29T^{2}
31 1−0.305T+31T2 1 - 0.305T + 31T^{2}
37 1+(−1.13−1.13i)T+37iT2 1 + (-1.13 - 1.13i)T + 37iT^{2}
41 1−4.48iT−41T2 1 - 4.48iT - 41T^{2}
43 1+(2.13−2.13i)T−43iT2 1 + (2.13 - 2.13i)T - 43iT^{2}
47 1+(−5.21+5.21i)T−47iT2 1 + (-5.21 + 5.21i)T - 47iT^{2}
53 1+(−9.69−9.69i)T+53iT2 1 + (-9.69 - 9.69i)T + 53iT^{2}
59 1−12.8T+59T2 1 - 12.8T + 59T^{2}
61 1+11.2T+61T2 1 + 11.2T + 61T^{2}
67 1+(1.52+1.52i)T+67iT2 1 + (1.52 + 1.52i)T + 67iT^{2}
71 1−11.3iT−71T2 1 - 11.3iT - 71T^{2}
73 1+(6.79−6.79i)T−73iT2 1 + (6.79 - 6.79i)T - 73iT^{2}
79 1+3.47iT−79T2 1 + 3.47iT - 79T^{2}
83 1+(5.49+5.49i)T+83iT2 1 + (5.49 + 5.49i)T + 83iT^{2}
89 1−9.03T+89T2 1 - 9.03T + 89T^{2}
97 1+(−0.595−0.595i)T+97iT2 1 + (-0.595 - 0.595i)T + 97iT^{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

−8.485685992456050363609839712687, −7.55615582193701603825370170018, −7.21488160653564471587758063479, −6.56950377280867725974703857891, −5.51177286665786341162834322295, −4.69800348252045193800422187456, −3.97519387870975053500252329560, −3.05240492065385069458847996656, −2.35306934480135335157885136738, −0.820702517071723870793153071009, 0.50066037140714451721306177754, 1.67146668663098332902751581373, 2.95926287282431679502083410851, 3.70451924088705379578779552941, 4.45439714977539397050222329743, 5.47041842242541578819776674764, 5.81974068028505288457016989573, 7.03043127070465534176002074954, 7.72873555461067549181075663879, 8.178932444086718854763645964489

Graph of the ZZ-function along the critical line