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Properties

Label 2-656-41.18-c1-0-14
Degree $2$
Conductor $656$
Sign $0.840 + 0.541i$
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  + 2.00·3-s + (1.06 − 0.770i)5-s + (0.823 − 2.53i)7-s + 1.02·9-s + (−0.383 − 0.278i)11-s + (0.477 + 1.46i)13-s + (2.12 − 1.54i)15-s + (4.19 + 3.04i)17-s + (1.37 − 4.24i)19-s + (1.65 − 5.09i)21-s + (−2.17 − 6.70i)23-s + (−1.01 + 3.12i)25-s − 3.95·27-s + (−0.481 + 0.349i)29-s + (2.56 + 1.86i)31-s + ⋯
L(s)  = 1  + 1.15·3-s + (0.474 − 0.344i)5-s + (0.311 − 0.958i)7-s + 0.343·9-s + (−0.115 − 0.0840i)11-s + (0.132 + 0.407i)13-s + (0.549 − 0.399i)15-s + (1.01 + 0.739i)17-s + (0.316 − 0.974i)19-s + (0.360 − 1.11i)21-s + (−0.454 − 1.39i)23-s + (−0.202 + 0.624i)25-s − 0.761·27-s + (−0.0893 + 0.0649i)29-s + (0.460 + 0.334i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(656\)    =    \(2^{4} \cdot 41\)
Sign: $0.840 + 0.541i$
Analytic conductor: \(5.23818\)
Root analytic conductor: \(2.28870\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{656} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 656,\ (\ :1/2),\ 0.840 + 0.541i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28009 - 0.671082i\)
\(L(\frac12)\) \(\approx\) \(2.28009 - 0.671082i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.33 - 3.53i)T \)
good3 \( 1 - 2.00T + 3T^{2} \)
5 \( 1 + (-1.06 + 0.770i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.823 + 2.53i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (0.383 + 0.278i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.477 - 1.46i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.19 - 3.04i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.37 + 4.24i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.17 + 6.70i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (0.481 - 0.349i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.56 - 1.86i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.06 + 0.772i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-2.10 - 6.48i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-0.404 - 1.24i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.60 + 1.16i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.42 - 7.47i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.08 + 6.40i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (9.25 - 6.72i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (3.24 + 2.35i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + 3.85T + 73T^{2} \)
79 \( 1 - 8.19T + 79T^{2} \)
83 \( 1 + 4.85T + 83T^{2} \)
89 \( 1 + (3.69 - 11.3i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-8.72 + 6.33i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.28391287379285322478477074939, −9.521242957239248182997579516517, −8.669205873725404800911395517990, −7.997295143386837046661620797071, −7.14346956535540778764047573814, −5.99345093472929161644191713173, −4.75290041157894613727901348233, −3.76772776395661816892321677162, −2.66775182325731964990032225277, −1.33473270280594034048871027614, 1.86200354267476556879475737439, 2.81619448932065264304496781361, 3.70741597715712085672526701938, 5.35106806459675107083428887504, 5.93896716722826970927431020586, 7.41755902102622946951815223112, 8.059340438532341258675105034117, 8.862150198989720743362890351595, 9.702333981710284031022303620905, 10.27884021570698515982823689285

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