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Properties

Label 2-416-104.69-c1-0-9
Degree $2$
Conductor $416$
Sign $-0.505 + 0.862i$
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  + (−2.36 − 1.36i)3-s + 0.267·5-s + (3 − 1.73i)7-s + (2.23 + 3.86i)9-s + (−1 + 1.73i)11-s + (2.59 − 2.5i)13-s + (−0.633 − 0.366i)15-s + (−3.23 − 5.59i)17-s + (−2.36 − 4.09i)19-s − 9.46·21-s + (−1.09 + 1.90i)23-s − 4.92·25-s − 4.00i·27-s + (−2.59 − 1.5i)29-s − 1.26i·31-s + ⋯
L(s)  = 1  + (−1.36 − 0.788i)3-s + 0.119·5-s + (1.13 − 0.654i)7-s + (0.744 + 1.28i)9-s + (−0.301 + 0.522i)11-s + (0.720 − 0.693i)13-s + (−0.163 − 0.0945i)15-s + (−0.783 − 1.35i)17-s + (−0.542 − 0.940i)19-s − 2.06·21-s + (−0.228 + 0.396i)23-s − 0.985·25-s − 0.769i·27-s + (−0.482 − 0.278i)29-s − 0.227i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.505 + 0.862i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.505 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.403594 - 0.704249i\)
\(L(\frac12)\) \(\approx\) \(0.403594 - 0.704249i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 + 2.5i)T \)
good3 \( 1 + (2.36 + 1.36i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.267T + 5T^{2} \)
7 \( 1 + (-3 + 1.73i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.23 + 5.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.36 + 4.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.09 - 1.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.59 + 1.5i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.26iT - 31T^{2} \)
37 \( 1 + (-3.86 + 6.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.03 + 0.598i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.19 + 4.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.26iT - 47T^{2} \)
53 \( 1 - 9.92iT - 53T^{2} \)
59 \( 1 + (3.73 + 6.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.36 - 9.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-11.0 + 6.36i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 - 5.46T + 83T^{2} \)
89 \( 1 + (0.464 + 0.267i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.19 + 3i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.17407340771169256239112235856, −10.41642076322190731101810053257, −9.090113357227083482947640547630, −7.69293831761107812313361325677, −7.26415426831664642076149403996, −6.10118608406956304866843771671, −5.21832158275791016213990045980, −4.31347696857637564898181324335, −2.10769982087713304484664145040, −0.63450756198110514803431586676, 1.78258460462793798500784011195, 3.95260671202757555823740954948, 4.76604116671770998357560017450, 5.89520505368743859118808443388, 6.25535885327853927750557500273, 8.031710309829496504434071972468, 8.773774087989420151178556101696, 9.980674843529552834752825905088, 10.89047503317491984597382740425, 11.27085798287056992531937664673

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