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Properties

Label 2-656-41.37-c1-0-6
Degree $2$
Conductor $656$
Sign $0.999 - 0.0431i$
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  − 1.59·3-s + (−0.571 + 1.75i)5-s + (−1.03 − 0.751i)7-s − 0.450·9-s + (−1.43 − 4.40i)11-s + (4.50 − 3.27i)13-s + (0.912 − 2.80i)15-s + (1.33 + 4.09i)17-s + (5.38 + 3.91i)19-s + (1.65 + 1.19i)21-s + (−2.53 + 1.84i)23-s + (1.27 + 0.928i)25-s + 5.50·27-s + (1.72 − 5.29i)29-s + (2.25 + 6.92i)31-s + ⋯
L(s)  = 1  − 0.921·3-s + (−0.255 + 0.786i)5-s + (−0.390 − 0.283i)7-s − 0.150·9-s + (−0.431 − 1.32i)11-s + (1.25 − 0.908i)13-s + (0.235 − 0.725i)15-s + (0.323 + 0.994i)17-s + (1.23 + 0.897i)19-s + (0.360 + 0.261i)21-s + (−0.528 + 0.383i)23-s + (0.255 + 0.185i)25-s + 1.06·27-s + (0.319 − 0.983i)29-s + (0.404 + 1.24i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $0.999 - 0.0431i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ 0.999 - 0.0431i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.948504 + 0.0204516i\)
\(L(\frac12)\) \(\approx\) \(0.948504 + 0.0204516i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.88 + 4.14i)T \)
good3 \( 1 + 1.59T + 3T^{2} \)
5 \( 1 + (0.571 - 1.75i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (1.03 + 0.751i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (1.43 + 4.40i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-4.50 + 3.27i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.33 - 4.09i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-5.38 - 3.91i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.53 - 1.84i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.72 + 5.29i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.25 - 6.92i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.60 - 4.94i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-6.69 + 4.86i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-1.44 + 1.05i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.27 + 6.98i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.72 + 2.70i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.91 - 3.56i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (1.10 - 3.41i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-1.34 - 4.12i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + 3.96T + 73T^{2} \)
79 \( 1 - 0.450T + 79T^{2} \)
83 \( 1 - 6.52T + 83T^{2} \)
89 \( 1 + (4.69 + 3.41i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.32 + 13.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.54535621026708244594554801211, −10.18807699416588167241733254455, −8.586091849768720081742639920051, −7.989952702616730733781155282320, −6.79885374045404535218425890661, −5.84622263739341381165619235693, −5.56079329826645137782639546288, −3.68955521412943861784307472919, −3.11947730741670372441692047178, −0.863538159458803055915782627921, 0.913791031569287057526341932107, 2.69582172794184237395498647355, 4.31377389061855262071588090917, 5.02900859521346434959528266950, 5.95322752724660669224767372932, 6.88290860718816427754935617588, 7.84485288989219995375822968146, 9.049373763989680572647374311400, 9.498108192905201206042718379114, 10.70773962859894549446381031190

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