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Properties

Label 2-656-41.33-c1-0-17
Degree $2$
Conductor $656$
Sign $0.0195 + 0.999i$
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  + (2.02 − 2.02i)3-s + (0.110 − 0.0358i)5-s + (0.422 − 2.66i)7-s − 5.19i·9-s + (1.53 + 3.01i)11-s + (−1.03 + 0.164i)13-s + (0.150 − 0.296i)15-s + (3.25 − 1.66i)17-s + (2.25 + 0.356i)19-s + (−4.54 − 6.25i)21-s + (−6.06 − 4.40i)23-s + (−4.03 + 2.93i)25-s + (−4.43 − 4.43i)27-s + (1.31 + 0.669i)29-s + (0.964 − 2.96i)31-s + ⋯
L(s)  = 1  + (1.16 − 1.16i)3-s + (0.0493 − 0.0160i)5-s + (0.159 − 1.00i)7-s − 1.73i·9-s + (0.463 + 0.909i)11-s + (−0.288 + 0.0456i)13-s + (0.0389 − 0.0764i)15-s + (0.790 − 0.402i)17-s + (0.516 + 0.0818i)19-s + (−0.991 − 1.36i)21-s + (−1.26 − 0.918i)23-s + (−0.806 + 0.586i)25-s + (−0.853 − 0.853i)27-s + (0.244 + 0.124i)29-s + (0.173 − 0.533i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0195 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0195 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $0.0195 + 0.999i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ 0.0195 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59498 - 1.56415i\)
\(L(\frac12)\) \(\approx\) \(1.59498 - 1.56415i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 - 6.26i)T \)
good3 \( 1 + (-2.02 + 2.02i)T - 3iT^{2} \)
5 \( 1 + (-0.110 + 0.0358i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-0.422 + 2.66i)T + (-6.65 - 2.16i)T^{2} \)
11 \( 1 + (-1.53 - 3.01i)T + (-6.46 + 8.89i)T^{2} \)
13 \( 1 + (1.03 - 0.164i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (-3.25 + 1.66i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (-2.25 - 0.356i)T + (18.0 + 5.87i)T^{2} \)
23 \( 1 + (6.06 + 4.40i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.31 - 0.669i)T + (17.0 + 23.4i)T^{2} \)
31 \( 1 + (-0.964 + 2.96i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.348 + 1.07i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (0.583 - 0.803i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (-1.73 - 10.9i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-0.482 - 0.246i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (-1.57 - 1.14i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (5.95 + 8.20i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (-6.02 + 11.8i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-4.28 - 8.40i)T + (-41.7 + 57.4i)T^{2} \)
73 \( 1 - 10.9iT - 73T^{2} \)
79 \( 1 + (7.58 - 7.58i)T - 79iT^{2} \)
83 \( 1 - 0.635T + 83T^{2} \)
89 \( 1 + (-0.753 + 4.75i)T + (-84.6 - 27.5i)T^{2} \)
97 \( 1 + (1.25 - 2.46i)T + (-57.0 - 78.4i)T^{2} \)
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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

−9.966076891668611163699534759033, −9.536743991132036519322120884359, −8.304345060681186041926627940787, −7.62498615999573036149582765487, −7.14008252988740086409139426409, −6.14165690329865369359488346302, −4.54421544691467048511750189003, −3.50881206571115325200934379080, −2.27852905691049271169835193346, −1.18217640787788950059666444217, 2.09709114451383202532952409151, 3.24298418694468152188643190292, 3.95539723791674611597851438899, 5.23759284862848822881721524820, 6.00292320699161323997459612902, 7.60942528377512234407546663483, 8.464986596607186427310702614051, 8.937882335782519252942162946549, 9.864039288384354294664481968926, 10.36208734056918601125914407288

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