Properties

Label 2-637-49.37-c1-0-32
Degree $2$
Conductor $637$
Sign $0.917 - 0.397i$
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  + (−1.79 + 1.22i)2-s + (1.34 − 1.24i)3-s + (0.991 − 2.52i)4-s + (1.96 + 0.607i)5-s + (−0.884 + 3.87i)6-s + (1.60 − 2.10i)7-s + (0.344 + 1.51i)8-s + (0.0264 − 0.353i)9-s + (−4.27 + 1.31i)10-s + (0.146 + 1.95i)11-s + (−1.81 − 4.62i)12-s + (0.900 + 0.433i)13-s + (−0.300 + 5.73i)14-s + (3.40 − 1.63i)15-s + (1.51 + 1.40i)16-s + (4.13 + 0.623i)17-s + ⋯
L(s)  = 1  + (−1.26 + 0.864i)2-s + (0.775 − 0.719i)3-s + (0.495 − 1.26i)4-s + (0.880 + 0.271i)5-s + (−0.361 + 1.58i)6-s + (0.605 − 0.795i)7-s + (0.121 + 0.534i)8-s + (0.00881 − 0.117i)9-s + (−1.35 + 0.417i)10-s + (0.0441 + 0.588i)11-s + (−0.524 − 1.33i)12-s + (0.249 + 0.120i)13-s + (−0.0802 + 1.53i)14-s + (0.877 − 0.422i)15-s + (0.377 + 0.350i)16-s + (1.00 + 0.151i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.917 - 0.397i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.917 - 0.397i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25970 + 0.260778i\)
\(L(\frac12)\) \(\approx\) \(1.25970 + 0.260778i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.60 + 2.10i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
good2 \( 1 + (1.79 - 1.22i)T + (0.730 - 1.86i)T^{2} \)
3 \( 1 + (-1.34 + 1.24i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (-1.96 - 0.607i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (-0.146 - 1.95i)T + (-10.8 + 1.63i)T^{2} \)
17 \( 1 + (-4.13 - 0.623i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (2.88 - 4.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.19 + 0.180i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (-4.84 + 6.07i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (5.35 + 9.27i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.147 - 0.376i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-2.35 - 10.3i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-0.267 + 1.17i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-10.7 + 7.35i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-1.56 + 3.97i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (8.61 - 2.65i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (2.15 + 5.48i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-3.49 - 6.04i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.28 - 2.85i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.0413 - 0.0281i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (6.04 - 10.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.62 - 4.15i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.575 - 7.68i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 - 0.774T + 97T^{2} \)
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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

−10.08139656442698907533087791477, −9.804773016483914455285868781779, −8.601473580897623987779602341183, −7.917670250644372188032065428938, −7.46718010199349945745458964834, −6.51873319719948989919435283434, −5.67729512649409452941842402370, −4.01849059405499945342473475400, −2.23505267824437288384961046115, −1.28940123569469301167854058962, 1.29187555643232272079814916241, 2.52127641060666391959593747148, 3.32724133048316571458324085303, 4.94323981827083382135407936106, 5.89792791937306398401492309011, 7.45264975674344763035718226221, 8.714518477225643525193587158358, 8.835873824915910867513781011928, 9.426623113118294858157075473852, 10.48778003554024948074961411807

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