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Properties

Label 2-656-41.25-c1-0-15
Degree $2$
Conductor $656$
Sign $-0.574 + 0.818i$
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  − 0.527i·3-s + (−0.0209 − 0.0152i)5-s + (−1.98 + 0.646i)7-s + 2.72·9-s + (−3.71 − 5.11i)11-s + (−4.33 − 1.40i)13-s + (−0.00801 + 0.0110i)15-s + (−1.06 − 1.46i)17-s + (2.80 − 0.912i)19-s + (0.340 + 1.04i)21-s + (−1.65 + 5.09i)23-s + (−1.54 − 4.75i)25-s − 3.01i·27-s + (1.98 − 2.72i)29-s + (7.58 − 5.51i)31-s + ⋯
L(s)  = 1  − 0.304i·3-s + (−0.00936 − 0.00680i)5-s + (−0.751 + 0.244i)7-s + 0.907·9-s + (−1.12 − 1.54i)11-s + (−1.20 − 0.390i)13-s + (−0.00206 + 0.00284i)15-s + (−0.257 − 0.354i)17-s + (0.644 − 0.209i)19-s + (0.0743 + 0.228i)21-s + (−0.345 + 1.06i)23-s + (−0.308 − 0.950i)25-s − 0.580i·27-s + (0.368 − 0.506i)29-s + (1.36 − 0.990i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $-0.574 + 0.818i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ -0.574 + 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.395617 - 0.761465i\)
\(L(\frac12)\) \(\approx\) \(0.395617 - 0.761465i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.82 + 2.66i)T \)
good3 \( 1 + 0.527iT - 3T^{2} \)
5 \( 1 + (0.0209 + 0.0152i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.98 - 0.646i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + (3.71 + 5.11i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (4.33 + 1.40i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.06 + 1.46i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.80 + 0.912i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.65 - 5.09i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-1.98 + 2.72i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-7.58 + 5.51i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (4.08 + 2.97i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (1.04 - 3.20i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (5.91 + 1.92i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.14 - 4.32i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.80 + 5.55i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.333 + 1.02i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-1.37 + 1.89i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-1.67 - 2.31i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 - 6.63iT - 79T^{2} \)
83 \( 1 + 2.63T + 83T^{2} \)
89 \( 1 + (4.03 - 1.31i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (6.82 - 9.38i)T + (-29.9 - 92.2i)T^{2} \)
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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

−9.964170509326630854110924747627, −9.704445185312495534845114151949, −8.288069280092153799971254729125, −7.69586527320641867944110722746, −6.67095998119399841494071077321, −5.75675628655404477797879522836, −4.79793474308716594818653493664, −3.36498613488640143565438166829, −2.42562906130422530115751635700, −0.43823437015706999107409834704, 1.91958005329854514756719524196, 3.21140798214192651864574292305, 4.57145376868406894272325134332, 5.03950778430627548286596141918, 6.70723928969492404737754521148, 7.14669821695770272957697556269, 8.136206317592828951037650110014, 9.441821120614315281607141492102, 10.18467006366928387382105015142, 10.26886821003193292322479238558

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