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Properties

Label 2-416-13.10-c1-0-4
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  + (1.45 + 2.51i)3-s + 2i·5-s + (0.675 + 0.389i)7-s + (−2.73 + 4.73i)9-s + (4.36 − 2.51i)11-s + (−1 − 3.46i)13-s + (−5.03 + 2.90i)15-s + (−3.23 + 5.59i)17-s + (0.675 + 0.389i)19-s + 2.26i·21-s + (−3.58 − 6.20i)23-s + 25-s − 7.16·27-s + (−1.5 − 2.59i)29-s − 1.55i·31-s + ⋯
L(s)  = 1  + (0.839 + 1.45i)3-s + 0.894i·5-s + (0.255 + 0.147i)7-s + (−0.910 + 1.57i)9-s + (1.31 − 0.759i)11-s + (−0.277 − 0.960i)13-s + (−1.30 + 0.751i)15-s + (−0.783 + 1.35i)17-s + (0.154 + 0.0894i)19-s + 0.494i·21-s + (−0.747 − 1.29i)23-s + 0.200·25-s − 1.37·27-s + (−0.278 − 0.482i)29-s − 0.280i·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

\(L(1)\) \(\approx\) \(1.11474 + 1.46244i\)
\(L(\frac12)\) \(\approx\) \(1.11474 + 1.46244i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.45 - 2.51i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2iT - 5T^{2} \)
7 \( 1 + (-0.675 - 0.389i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.36 + 2.51i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.23 - 5.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.675 - 0.389i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.58 + 6.20i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.55iT - 31T^{2} \)
37 \( 1 + (3.69 - 2.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.76 - 1.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.36 + 7.55i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.37iT - 47T^{2} \)
53 \( 1 - 9.46T + 53T^{2} \)
59 \( 1 + (-10.7 - 6.20i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.675 - 0.389i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-13.0 - 7.55i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + 10.0iT - 83T^{2} \)
89 \( 1 + (7.96 - 4.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.89 + 5.13i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.03705323814994831204979774996, −10.53683893021585835607344477068, −9.779135736361888305910173115929, −8.683724067135507555557968766374, −8.262306326487126236312740801790, −6.73629759756520469468812598849, −5.67672346210766118670216921156, −4.25286770059819553037975205816, −3.57978343572820033313778666103, −2.46052767009696544430867724731, 1.25549760220754058442435390340, 2.19619225988201727065988138706, 3.87742847627620100571773913165, 5.09192932667874622684434748665, 6.72369427495300349500842348114, 7.11394792244856301572318502555, 8.160718122916270441875863541282, 9.165183020454601541233588590107, 9.435518668552200857355012294545, 11.41573353292577166211613175843

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