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Properties

Label 2-416-52.31-c1-0-7
Degree $2$
Conductor $416$
Sign $0.916 - 0.399i$
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  + i·3-s + (1.58 − 1.58i)5-s + (−1.58 + 1.58i)7-s + 2·9-s + (2.16 − 2.16i)11-s + (0.418 + 3.58i)13-s + (1.58 + 1.58i)15-s − 5.32i·17-s + (5.16 + 5.16i)19-s + (−1.58 − 1.58i)21-s − 0.837·23-s + 5i·27-s + 5.16·29-s + (−5.16 − 5.16i)31-s + (2.16 + 2.16i)33-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.707 − 0.707i)5-s + (−0.597 + 0.597i)7-s + 0.666·9-s + (0.651 − 0.651i)11-s + (0.116 + 0.993i)13-s + (0.408 + 0.408i)15-s − 1.29i·17-s + (1.18 + 1.18i)19-s + (−0.345 − 0.345i)21-s − 0.174·23-s + 0.962i·27-s + 0.958·29-s + (−0.927 − 0.927i)31-s + (0.376 + 0.376i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.916 - 0.399i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.916 - 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56777 + 0.326328i\)
\(L(\frac12)\) \(\approx\) \(1.56777 + 0.326328i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.418 - 3.58i)T \)
good3 \( 1 - iT - 3T^{2} \)
5 \( 1 + (-1.58 + 1.58i)T - 5iT^{2} \)
7 \( 1 + (1.58 - 1.58i)T - 7iT^{2} \)
11 \( 1 + (-2.16 + 2.16i)T - 11iT^{2} \)
17 \( 1 + 5.32iT - 17T^{2} \)
19 \( 1 + (-5.16 - 5.16i)T + 19iT^{2} \)
23 \( 1 + 0.837T + 23T^{2} \)
29 \( 1 - 5.16T + 29T^{2} \)
31 \( 1 + (5.16 + 5.16i)T + 31iT^{2} \)
37 \( 1 + (0.418 + 0.418i)T + 37iT^{2} \)
41 \( 1 + (1.16 - 1.16i)T - 41iT^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (2.74 - 2.74i)T - 47iT^{2} \)
53 \( 1 + 9.48T + 53T^{2} \)
59 \( 1 + (-4 + 4i)T - 59iT^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (5.32 + 5.32i)T + 67iT^{2} \)
71 \( 1 + (1.58 + 1.58i)T + 71iT^{2} \)
73 \( 1 + (6 + 6i)T + 73iT^{2} \)
79 \( 1 + 15.4iT - 79T^{2} \)
83 \( 1 + (12.1 + 12.1i)T + 83iT^{2} \)
89 \( 1 + (9.16 + 9.16i)T + 89iT^{2} \)
97 \( 1 + (10.1 - 10.1i)T - 97iT^{2} \)
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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

−11.36609628341831555198996475285, −10.04673314944775013351680343469, −9.335100372518998475114935465614, −9.070354792375638400820064616820, −7.56998860042287145816072372559, −6.36276341203612987025806853011, −5.49802846498122549012620101688, −4.45107004662443771485826823514, −3.24578089418691820703278293331, −1.52146162835706676448685326747, 1.35803687814583713231729708358, 2.84829332137229178515515376937, 4.10682769139053428077270907473, 5.60417136958322798776392147893, 6.79261164471512480903367542762, 7.02573440585342310792948446174, 8.290661586965020514607531681157, 9.650639200467231456222435574709, 10.16074487840576812857458011226, 10.96908922034121466382222890530

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