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Properties

Label 2-416-32.21-c1-0-12
Degree $2$
Conductor $416$
Sign $0.428 - 0.903i$
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  + (−1.40 + 0.152i)2-s + (−0.435 + 0.180i)3-s + (1.95 − 0.427i)4-s + (0.326 − 0.788i)5-s + (0.585 − 0.320i)6-s + (1.89 + 1.89i)7-s + (−2.68 + 0.898i)8-s + (−1.96 + 1.96i)9-s + (−0.339 + 1.15i)10-s + (1.20 + 0.500i)11-s + (−0.774 + 0.539i)12-s + (−0.382 − 0.923i)13-s + (−2.94 − 2.37i)14-s + 0.402i·15-s + (3.63 − 1.67i)16-s + 1.89i·17-s + ⋯
L(s)  = 1  + (−0.994 + 0.107i)2-s + (−0.251 + 0.104i)3-s + (0.976 − 0.213i)4-s + (0.145 − 0.352i)5-s + (0.239 − 0.130i)6-s + (0.714 + 0.714i)7-s + (−0.948 + 0.317i)8-s + (−0.654 + 0.654i)9-s + (−0.107 + 0.366i)10-s + (0.364 + 0.150i)11-s + (−0.223 + 0.155i)12-s + (−0.106 − 0.256i)13-s + (−0.787 − 0.633i)14-s + 0.103i·15-s + (0.908 − 0.418i)16-s + 0.458i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.428 - 0.903i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.428 - 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.699118 + 0.442272i\)
\(L(\frac12)\) \(\approx\) \(0.699118 + 0.442272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.152i)T \)
13 \( 1 + (0.382 + 0.923i)T \)
good3 \( 1 + (0.435 - 0.180i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.326 + 0.788i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.89 - 1.89i)T + 7iT^{2} \)
11 \( 1 + (-1.20 - 0.500i)T + (7.77 + 7.77i)T^{2} \)
17 \( 1 - 1.89iT - 17T^{2} \)
19 \( 1 + (-0.478 - 1.15i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.331 - 0.331i)T - 23iT^{2} \)
29 \( 1 + (-0.842 + 0.349i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 + (4.38 - 10.5i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (8.18 - 8.18i)T - 41iT^{2} \)
43 \( 1 + (-9.38 - 3.88i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 4.42iT - 47T^{2} \)
53 \( 1 + (-1.51 - 0.626i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-4.71 + 11.3i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-10.2 + 4.24i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (0.969 - 0.401i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-5.81 - 5.81i)T + 71iT^{2} \)
73 \( 1 + (6.10 - 6.10i)T - 73iT^{2} \)
79 \( 1 + 4.79iT - 79T^{2} \)
83 \( 1 + (4.15 + 10.0i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (2.78 + 2.78i)T + 89iT^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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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

−11.38370985744105637745517607076, −10.38026376546088921599675710216, −9.544490982868037715850084926833, −8.358170603403198829930780284108, −8.213117755029727202933506287311, −6.75265325588550515244401823486, −5.71479160467744957991612749910, −4.87472708556401376884796608735, −2.85184991645459444407365153179, −1.52356058045961887022914078193, 0.814271679901212791286990174948, 2.47994573903582241138665026070, 3.87956793228244409625793902292, 5.51330234675057906639890013357, 6.68395375954733265753819140551, 7.27278892799193795072177567565, 8.472931262418226755235090412067, 9.135031652983562891653327181924, 10.28536411718911042932795765500, 10.90348539093745912603836948751

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