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Properties

Label 2-637-13.12-c1-0-21
Degree $2$
Conductor $637$
Sign $0.235 - 0.971i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
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  + 1.07i·2-s + 2.43·3-s + 0.848·4-s + 0.625i·5-s + 2.60i·6-s + 3.05i·8-s + 2.91·9-s − 0.671·10-s − 0.708i·11-s + 2.06·12-s + (−0.848 + 3.50i)13-s + 1.52i·15-s − 1.58·16-s − 3.34·17-s + 3.12i·18-s − 5.20i·19-s + ⋯
L(s)  = 1  + 0.758i·2-s + 1.40·3-s + 0.424·4-s + 0.279i·5-s + 1.06i·6-s + 1.08i·8-s + 0.970·9-s − 0.212·10-s − 0.213i·11-s + 0.595·12-s + (−0.235 + 0.971i)13-s + 0.392i·15-s − 0.395·16-s − 0.810·17-s + 0.736i·18-s − 1.19i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.235 - 0.971i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.235 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07917 + 1.63579i\)
\(L(\frac12)\) \(\approx\) \(2.07917 + 1.63579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (0.848 - 3.50i)T \)
good2 \( 1 - 1.07iT - 2T^{2} \)
3 \( 1 - 2.43T + 3T^{2} \)
5 \( 1 - 0.625iT - 5T^{2} \)
11 \( 1 + 0.708iT - 11T^{2} \)
17 \( 1 + 3.34T + 17T^{2} \)
19 \( 1 + 5.20iT - 19T^{2} \)
23 \( 1 - 4.43T + 23T^{2} \)
29 \( 1 + 6.59T + 29T^{2} \)
31 \( 1 + 4.39iT - 31T^{2} \)
37 \( 1 - 0.423iT - 37T^{2} \)
41 \( 1 + 5.01iT - 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + 8.07iT - 47T^{2} \)
53 \( 1 + 0.697T + 53T^{2} \)
59 \( 1 + 9.86iT - 59T^{2} \)
61 \( 1 + 4.69T + 61T^{2} \)
67 \( 1 + 10.4iT - 67T^{2} \)
71 \( 1 - 14.0iT - 71T^{2} \)
73 \( 1 - 5.08iT - 73T^{2} \)
79 \( 1 + 3.91T + 79T^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 - 13.3iT - 89T^{2} \)
97 \( 1 + 0.202iT - 97T^{2} \)
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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

−10.93408468896849472068172031619, −9.399025168478534472812673211226, −8.959545568294475396428782543758, −8.103260965922860194802805695852, −7.12622688245694231937623407562, −6.76017764275034811127740865699, −5.39103264744931981573343242117, −4.14322769692572367062881787866, −2.84145009302106678454844846903, −2.11252940288577317462382971826, 1.45310485465070194576287702033, 2.62855738600029369640628003850, 3.30664279879644551127185781176, 4.40284294805630125656501791526, 5.87535913472554744672778701638, 7.17787821219286752195765662801, 7.82143965375371956111850360680, 8.852355383255991748143467988030, 9.472006302570746412427827341469, 10.43261540019428047818063568035

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