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Properties

Label 2-416-13.10-c1-0-3
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  + (0.866 + 1.5i)3-s + 3.46i·5-s + (2.59 + 1.5i)7-s + (−4.33 + 2.5i)11-s + (−1 − 3.46i)13-s + (−5.19 + 2.99i)15-s + (3.5 − 6.06i)17-s + (−4.33 − 2.5i)19-s + 5.19i·21-s + (2.59 + 4.5i)23-s − 6.99·25-s + 5.19·27-s + (2.5 + 4.33i)29-s + 2i·31-s + (−7.5 − 4.33i)33-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)3-s + 1.54i·5-s + (0.981 + 0.566i)7-s + (−1.30 + 0.753i)11-s + (−0.277 − 0.960i)13-s + (−1.34 + 0.774i)15-s + (0.848 − 1.47i)17-s + (−0.993 − 0.573i)19-s + 1.13i·21-s + (0.541 + 0.938i)23-s − 1.39·25-s + 1.00·27-s + (0.464 + 0.804i)29-s + 0.359i·31-s + (−1.30 − 0.753i)33-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

\(L(1)\) \(\approx\) \(1.00432 + 1.31759i\)
\(L(\frac12)\) \(\approx\) \(1.00432 + 1.31759i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 1.5i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 + (-2.59 - 1.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.33 - 2.5i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.5 + 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.33 + 2.5i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.59 - 4.5i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 + (-4.5 + 2.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.59 + 4.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (-6.06 - 3.5i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.59 - 1.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.06 + 3.5i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.46iT - 73T^{2} \)
79 \( 1 + 3.46T + 79T^{2} \)
83 \( 1 + 14iT - 83T^{2} \)
89 \( 1 + (-1.5 + 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.5 - 4.33i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.18730186715297796826659150158, −10.45289529232732521623226162730, −9.922392955102943269953430468258, −8.830804606247835746766407911564, −7.67431865783985467018022019400, −7.09480649625657389865806807161, −5.51169308262128402487868568562, −4.69551890972061568647975851003, −3.13280708837345839580850573153, −2.54894164571844234500184399434, 1.11471873287324465725549016299, 2.20001653520271263739376382071, 4.19300724258586834100372526043, 4.99253488121029589828855222896, 6.18570872055557231223196832515, 7.65184186003859798424435421020, 8.252334559306368302886744114880, 8.592667418902307145411061986442, 10.08894683545246689525724414725, 10.94908538175983714923386239358

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