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Properties

Label 2-416-13.10-c1-0-10
Degree $2$
Conductor $416$
Sign $0.265 + 0.964i$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.619 − 1.07i)3-s − 2i·5-s + (4.00 + 2.31i)7-s + (0.732 − 1.26i)9-s + (−1.85 + 1.07i)11-s + (−1 − 3.46i)13-s + (−2.14 + 1.23i)15-s + (0.232 − 0.401i)17-s + (4.00 + 2.31i)19-s − 5.73i·21-s + (−2.76 − 4.79i)23-s + 25-s − 5.53·27-s + (−1.5 − 2.59i)29-s − 9.25i·31-s + ⋯
L(s)  = 1  + (−0.357 − 0.619i)3-s − 0.894i·5-s + (1.51 + 0.874i)7-s + (0.244 − 0.422i)9-s + (−0.560 + 0.323i)11-s + (−0.277 − 0.960i)13-s + (−0.554 + 0.319i)15-s + (0.0562 − 0.0974i)17-s + (0.918 + 0.530i)19-s − 1.25i·21-s + (−0.576 − 0.999i)23-s + 0.200·25-s − 1.06·27-s + (−0.278 − 0.482i)29-s − 1.66i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.265 + 0.964i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.265 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09835 - 0.837218i\)
\(L(\frac12)\) \(\approx\) \(1.09835 - 0.837218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (1 + 3.46i)T \)
good3 \( 1 + (0.619 + 1.07i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 + (-4.00 - 2.31i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.85 - 1.07i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.232 + 0.401i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.00 - 2.31i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.76 + 4.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.25iT - 31T^{2} \)
37 \( 1 + (-6.69 + 3.86i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.23 - 3.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.85 - 3.21i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.7iT - 47T^{2} \)
53 \( 1 - 2.53T + 53T^{2} \)
59 \( 1 + (-8.29 - 4.79i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.00 - 2.31i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.57 + 3.21i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 4.29T + 79T^{2} \)
83 \( 1 - 4.29iT - 83T^{2} \)
89 \( 1 + (1.03 - 0.598i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.8 - 6.86i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.35408886688630642585445331334, −10.08806398786797250892964511995, −9.113259503956614243682656895656, −7.990518267979334447046074562532, −7.69664714416624774970310076032, −6.03428051545010489667513317975, −5.30890435539017129260970331786, −4.38956819989534543368259184009, −2.39144556708733591034295391988, −1.05325020756801103424235098021, 1.79651805371411852427655471626, 3.50126727094041095816937563672, 4.69965266893849253746246853253, 5.33040217324317127803146178408, 6.97660760018063334746861857320, 7.52225638566090454891065034108, 8.603453937559120483965508278068, 9.982815536599453991414719852995, 10.55134966998982832198447014208, 11.27457946594644196168254246240

Graph of the $Z$-function along the critical line