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Properties

Label 2-656-656.245-c1-0-3
Degree $2$
Conductor $656$
Sign $-0.890 + 0.455i$
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  + (−1.41 + 0.0856i)2-s + (2.26 + 2.26i)3-s + (1.98 − 0.241i)4-s + (−1.97 − 1.97i)5-s + (−3.38 − 3.00i)6-s − 2.05·7-s + (−2.78 + 0.511i)8-s + 7.23i·9-s + (2.95 + 2.61i)10-s + (−3.02 + 3.02i)11-s + (5.03 + 3.94i)12-s + (−3.75 − 3.75i)13-s + (2.89 − 0.175i)14-s − 8.92i·15-s + (3.88 − 0.960i)16-s − 1.85i·17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0605i)2-s + (1.30 + 1.30i)3-s + (0.992 − 0.120i)4-s + (−0.882 − 0.882i)5-s + (−1.38 − 1.22i)6-s − 0.774·7-s + (−0.983 + 0.180i)8-s + 2.41i·9-s + (0.934 + 0.827i)10-s + (−0.912 + 0.912i)11-s + (1.45 + 1.13i)12-s + (−1.04 − 1.04i)13-s + (0.773 − 0.0469i)14-s − 2.30i·15-s + (0.970 − 0.240i)16-s − 0.450i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $-0.890 + 0.455i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ -0.890 + 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0506486 - 0.210043i\)
\(L(\frac12)\) \(\approx\) \(0.0506486 - 0.210043i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0856i)T \)
41 \( 1 + (5.19 - 3.73i)T \)
good3 \( 1 + (-2.26 - 2.26i)T + 3iT^{2} \)
5 \( 1 + (1.97 + 1.97i)T + 5iT^{2} \)
7 \( 1 + 2.05T + 7T^{2} \)
11 \( 1 + (3.02 - 3.02i)T - 11iT^{2} \)
13 \( 1 + (3.75 + 3.75i)T + 13iT^{2} \)
17 \( 1 + 1.85iT - 17T^{2} \)
19 \( 1 + (4.54 + 4.54i)T + 19iT^{2} \)
23 \( 1 - 8.60iT - 23T^{2} \)
29 \( 1 + (0.660 + 0.660i)T + 29iT^{2} \)
31 \( 1 - 4.09T + 31T^{2} \)
37 \( 1 + (4.46 + 4.46i)T + 37iT^{2} \)
43 \( 1 + (-3.31 - 3.31i)T + 43iT^{2} \)
47 \( 1 - 4.72iT - 47T^{2} \)
53 \( 1 + (-2.85 + 2.85i)T - 53iT^{2} \)
59 \( 1 + (-6.34 - 6.34i)T + 59iT^{2} \)
61 \( 1 + (0.395 - 0.395i)T - 61iT^{2} \)
67 \( 1 + (-1.51 - 1.51i)T + 67iT^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 - 4.39iT - 73T^{2} \)
79 \( 1 + 4.14iT - 79T^{2} \)
83 \( 1 + (0.624 - 0.624i)T - 83iT^{2} \)
89 \( 1 - 5.65T + 89T^{2} \)
97 \( 1 + 2.71iT - 97T^{2} \)
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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

−10.50391334944159244310030995720, −9.920020582336407876432156639567, −9.347897443107912355299598011629, −8.576308122323242300431854336874, −7.81482704304936687390623615305, −7.26896164806992057233075225735, −5.31715868535809128808597292404, −4.49917563091571489297200123797, −3.23811850037634767046325371105, −2.43229864169566539875715324863, 0.12183229779488712590948865917, 2.10371960308334327938731497800, 2.86782944035572685610444892202, 3.72115404897982797137090598095, 6.32945803428671329051981681408, 6.78504890767316683441124461102, 7.54379088761626631203485441202, 8.340917699864671172121624420653, 8.743754993645913828974531826622, 9.994042159150978491374294722704

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