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Properties

Label 2-656-41.37-c1-0-19
Degree $2$
Conductor $656$
Sign $-0.846 + 0.532i$
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  + 0.618·3-s + (0.5 − 1.53i)5-s + (−2.11 − 1.53i)7-s − 2.61·9-s + (−0.927 − 2.85i)11-s + (−5.04 + 3.66i)13-s + (0.309 − 0.951i)15-s + (−1.30 − 4.02i)17-s + (−1.42 − 1.03i)19-s + (−1.30 − 0.951i)21-s + (−1.30 + 0.951i)23-s + (1.92 + 1.40i)25-s − 3.47·27-s + (−0.163 + 0.502i)29-s + (1.88 + 5.79i)31-s + ⋯
L(s)  = 1  + 0.356·3-s + (0.223 − 0.688i)5-s + (−0.800 − 0.581i)7-s − 0.872·9-s + (−0.279 − 0.860i)11-s + (−1.39 + 1.01i)13-s + (0.0797 − 0.245i)15-s + (−0.317 − 0.977i)17-s + (−0.327 − 0.237i)19-s + (−0.285 − 0.207i)21-s + (−0.272 + 0.198i)23-s + (0.385 + 0.280i)25-s − 0.668·27-s + (−0.0302 + 0.0932i)29-s + (0.338 + 1.04i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $-0.846 + 0.532i$
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: \((2,\ 656,\ (\ :1/2),\ -0.846 + 0.532i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.198126 - 0.687077i\)
\(L(\frac12)\) \(\approx\) \(0.198126 - 0.687077i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 + 6.01i)T \)
good3 \( 1 - 0.618T + 3T^{2} \)
5 \( 1 + (-0.5 + 1.53i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.11 + 1.53i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (0.927 + 2.85i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (5.04 - 3.66i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.30 + 4.02i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.42 + 1.03i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.30 - 0.951i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.163 - 0.502i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.88 - 5.79i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.454 + 1.40i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-7.85 + 5.70i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (1.23 - 0.898i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.66 + 11.2i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (4.11 - 2.99i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.80 - 2.04i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-4.76 + 14.6i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.19 + 3.66i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + 16.1T + 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 - 16.0T + 83T^{2} \)
89 \( 1 + (-5.04 - 3.66i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.69 + 5.20i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.03012235602856444129020250693, −9.144277665720754240025190971444, −8.751753305470974191666508667332, −7.49618144751831945385636116508, −6.71924507562127281698122708415, −5.55304768031695424281329528275, −4.66548591365412410503772717515, −3.38587787620042086336912469204, −2.33799637269863992992477029222, −0.33305462174410690175554500507, 2.47235359717407633437703381890, 2.84549539760076544214068257938, 4.36230293429045609439605358518, 5.68105442655202632484546209493, 6.33385835615241441413053247641, 7.47172022832058707518111494583, 8.225040895131857858780139775043, 9.305752928460447467533371408710, 10.01403180313552472472262940417, 10.66346845852499884609828263822

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