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Properties

Label 2-656-656.245-c1-0-64
Degree $2$
Conductor $656$
Sign $-0.342 + 0.939i$
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  + (−1.23 + 0.686i)2-s + (0.0704 + 0.0704i)3-s + (1.05 − 1.69i)4-s + (−2.73 − 2.73i)5-s + (−0.135 − 0.0386i)6-s + 4.71·7-s + (−0.140 + 2.82i)8-s − 2.99i·9-s + (5.26 + 1.50i)10-s + (−1.43 + 1.43i)11-s + (0.193 − 0.0451i)12-s + (−2.35 − 2.35i)13-s + (−5.82 + 3.23i)14-s − 0.385i·15-s + (−1.76 − 3.58i)16-s − 3.04i·17-s + ⋯
L(s)  = 1  + (−0.874 + 0.485i)2-s + (0.0406 + 0.0406i)3-s + (0.528 − 0.849i)4-s + (−1.22 − 1.22i)5-s + (−0.0552 − 0.0157i)6-s + 1.78·7-s + (−0.0495 + 0.998i)8-s − 0.996i·9-s + (1.66 + 0.475i)10-s + (−0.433 + 0.433i)11-s + (0.0559 − 0.0130i)12-s + (−0.654 − 0.654i)13-s + (−1.55 + 0.864i)14-s − 0.0995i·15-s + (−0.441 − 0.897i)16-s − 0.738i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $-0.342 + 0.939i$
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: \((2,\ 656,\ (\ :1/2),\ -0.342 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.380209 - 0.543503i\)
\(L(\frac12)\) \(\approx\) \(0.380209 - 0.543503i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 - 0.686i)T \)
41 \( 1 + (5.86 + 2.58i)T \)
good3 \( 1 + (-0.0704 - 0.0704i)T + 3iT^{2} \)
5 \( 1 + (2.73 + 2.73i)T + 5iT^{2} \)
7 \( 1 - 4.71T + 7T^{2} \)
11 \( 1 + (1.43 - 1.43i)T - 11iT^{2} \)
13 \( 1 + (2.35 + 2.35i)T + 13iT^{2} \)
17 \( 1 + 3.04iT - 17T^{2} \)
19 \( 1 + (-3.59 - 3.59i)T + 19iT^{2} \)
23 \( 1 - 3.02iT - 23T^{2} \)
29 \( 1 + (6.96 + 6.96i)T + 29iT^{2} \)
31 \( 1 + 5.91T + 31T^{2} \)
37 \( 1 + (1.93 + 1.93i)T + 37iT^{2} \)
43 \( 1 + (3.26 + 3.26i)T + 43iT^{2} \)
47 \( 1 + 1.15iT - 47T^{2} \)
53 \( 1 + (-7.39 + 7.39i)T - 53iT^{2} \)
59 \( 1 + (-4.01 - 4.01i)T + 59iT^{2} \)
61 \( 1 + (-0.0997 + 0.0997i)T - 61iT^{2} \)
67 \( 1 + (8.32 + 8.32i)T + 67iT^{2} \)
71 \( 1 - 8.28T + 71T^{2} \)
73 \( 1 + 7.24iT - 73T^{2} \)
79 \( 1 + 1.97iT - 79T^{2} \)
83 \( 1 + (6.50 - 6.50i)T - 83iT^{2} \)
89 \( 1 - 6.92T + 89T^{2} \)
97 \( 1 - 11.9iT - 97T^{2} \)
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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

−10.05499007236088732437088089217, −9.172732831183487526361751883323, −8.435572709542097829566428687412, −7.60006336117728724317212306766, −7.43856558847694450652427750473, −5.43285757683082152412799507810, −5.07377401926301101546574252751, −3.82816693425914918332034652466, −1.76848560352709998307842625954, −0.47883203012899490832009664756, 1.80103545153657370706055547064, 2.87893642018311044749724869324, 4.08433194630733107078792948644, 5.15786324078878601879326361951, 7.00307612829397117166962407763, 7.48691075539446861919114592910, 8.118081147966559466964546684523, 8.838065052800557171443112524097, 10.34629424470491709557864432931, 10.89351617043830608975191871859

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