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Properties

Label 2-637-13.3-c1-0-40
Degree $2$
Conductor $637$
Sign $-0.0910 - 0.995i$
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  + (−0.5 − 0.866i)2-s + (−0.707 − 1.22i)3-s + (0.500 − 0.866i)4-s − 2.68·5-s + (−0.707 + 1.22i)6-s − 3·8-s + (0.500 − 0.866i)9-s + (1.34 + 2.32i)10-s + (−2.89 − 5.01i)11-s − 1.41·12-s + (2.75 + 2.32i)13-s + (1.89 + 3.28i)15-s + (0.500 + 0.866i)16-s + (−2.75 + 4.77i)17-s − 1.00·18-s + (1.41 − 2.44i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.408 − 0.707i)3-s + (0.250 − 0.433i)4-s − 1.20·5-s + (−0.288 + 0.499i)6-s − 1.06·8-s + (0.166 − 0.288i)9-s + (0.424 + 0.735i)10-s + (−0.873 − 1.51i)11-s − 0.408·12-s + (0.764 + 0.644i)13-s + (0.490 + 0.848i)15-s + (0.125 + 0.216i)16-s + (−0.668 + 1.15i)17-s − 0.235·18-s + (0.324 − 0.561i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0910 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0910 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.0910 - 0.995i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.0910 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.188615 + 0.206650i\)
\(L(\frac12)\) \(\approx\) \(0.188615 + 0.206650i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.75 - 2.32i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.707 + 1.22i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 2.68T + 5T^{2} \)
11 \( 1 + (2.89 + 5.01i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.75 - 4.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.897 + 1.55i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.39 - 7.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.41T + 31T^{2} \)
37 \( 1 + (-3.39 - 5.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.87 + 8.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.897 + 1.55i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 6.59T + 53T^{2} \)
59 \( 1 + (0.562 - 0.974i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.779 - 1.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.89 + 5.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.80T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 + (6.07 + 10.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.12 + 3.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.32353859423880609255438958361, −8.824733368849370840523887390814, −8.456570652312614467993370018215, −7.18847970187614787559965052292, −6.40226118000252867285341350428, −5.58581701752098376449706748912, −4.01453366613151528631089594806, −2.99003773010858624601410018975, −1.38600325544799745284109894002, −0.18581199578397290421494502076, 2.61430168286375432095907888604, 3.93042181009860201554773652978, 4.71437182543643194575479475513, 5.82776905307783662093603566839, 7.15888991704655362040034975412, 7.66567886378076112982520498919, 8.296496958172166040918269324395, 9.531854006169757356907107166364, 10.27064009161263155873701713512, 11.30380723252950505791218987696

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