Properties

Label 2-656-41.5-c1-0-14
Degree $2$
Conductor $656$
Sign $0.881 + 0.472i$
Analytic cond. $5.23818$
Root an. cond. $2.28870$
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  + (0.983 + 0.983i)3-s + (1.02 + 0.331i)5-s + (−0.625 − 3.95i)7-s − 1.06i·9-s + (1.14 − 2.23i)11-s + (4.25 + 0.673i)13-s + (0.677 + 1.33i)15-s + (−4.26 − 2.17i)17-s + (−0.967 + 0.153i)19-s + (3.27 − 4.50i)21-s + (1.78 − 1.29i)23-s + (−3.11 − 2.26i)25-s + (3.99 − 3.99i)27-s + (1.83 − 0.933i)29-s + (2.02 + 6.23i)31-s + ⋯
L(s)  = 1  + (0.567 + 0.567i)3-s + (0.456 + 0.148i)5-s + (−0.236 − 1.49i)7-s − 0.355i·9-s + (0.344 − 0.675i)11-s + (1.17 + 0.186i)13-s + (0.174 + 0.343i)15-s + (−1.03 − 0.526i)17-s + (−0.221 + 0.0351i)19-s + (0.713 − 0.982i)21-s + (0.372 − 0.270i)23-s + (−0.622 − 0.452i)25-s + (0.769 − 0.769i)27-s + (0.340 − 0.173i)29-s + (0.363 + 1.12i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $0.881 + 0.472i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ 0.881 + 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86149 - 0.467527i\)
\(L(\frac12)\) \(\approx\) \(1.86149 - 0.467527i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 + (-6.18 - 1.67i)T \)
good3 \( 1 + (-0.983 - 0.983i)T + 3iT^{2} \)
5 \( 1 + (-1.02 - 0.331i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.625 + 3.95i)T + (-6.65 + 2.16i)T^{2} \)
11 \( 1 + (-1.14 + 2.23i)T + (-6.46 - 8.89i)T^{2} \)
13 \( 1 + (-4.25 - 0.673i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (4.26 + 2.17i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (0.967 - 0.153i)T + (18.0 - 5.87i)T^{2} \)
23 \( 1 + (-1.78 + 1.29i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.83 + 0.933i)T + (17.0 - 23.4i)T^{2} \)
31 \( 1 + (-2.02 - 6.23i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.34 - 10.3i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-1.18 - 1.63i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (1.55 - 9.83i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-6.91 + 3.52i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (7.77 - 5.64i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.263 - 0.362i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.59 - 3.12i)T + (-39.3 + 54.2i)T^{2} \)
71 \( 1 + (-6.39 + 12.5i)T + (-41.7 - 57.4i)T^{2} \)
73 \( 1 + 8.75iT - 73T^{2} \)
79 \( 1 + (-1.93 - 1.93i)T + 79iT^{2} \)
83 \( 1 - 2.79T + 83T^{2} \)
89 \( 1 + (-1.73 - 10.9i)T + (-84.6 + 27.5i)T^{2} \)
97 \( 1 + (-1.70 - 3.35i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.49672936470888164931529418462, −9.578486494553836554969640748735, −8.873440015128510403531053964105, −8.025415196216526591255264210777, −6.65834492218646753771388605188, −6.32578314246535922863648730777, −4.60994591575646103141719424617, −3.85224585854450258823573047991, −2.95323158610769649682376262728, −1.06688329638829664013633884533, 1.82235792690617575187118507051, 2.48094624319306155346829118135, 3.94133006769566700398509579950, 5.36231752796674856183794536395, 6.08683534622271286554108589460, 7.07037946188877526622141225026, 8.196689891860180752061401867430, 8.865944672061081532135119884347, 9.413202759737568634426485901679, 10.64384173677824997906122589207

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