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Properties

Label 2-637-7.4-c1-0-3
Degree $2$
Conductor $637$
Sign $-0.900 - 0.435i$
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  + (−0.132 − 0.229i)2-s + (−1.45 + 2.51i)3-s + (0.964 − 1.67i)4-s + (0.717 + 1.24i)5-s + 0.769·6-s − 1.03·8-s + (−2.73 − 4.73i)9-s + (0.189 − 0.328i)10-s + (−2.75 + 4.76i)11-s + (2.80 + 4.86i)12-s + 13-s − 4.17·15-s + (−1.79 − 3.10i)16-s + (−2.41 + 4.18i)17-s + (−0.722 + 1.25i)18-s + (−1.41 − 2.44i)19-s + ⋯
L(s)  = 1  + (−0.0935 − 0.162i)2-s + (−0.839 + 1.45i)3-s + (0.482 − 0.835i)4-s + (0.320 + 0.555i)5-s + 0.314·6-s − 0.367·8-s + (−0.910 − 1.57i)9-s + (0.0600 − 0.104i)10-s + (−0.829 + 1.43i)11-s + (0.810 + 1.40i)12-s + 0.277·13-s − 1.07·15-s + (−0.448 − 0.776i)16-s + (−0.585 + 1.01i)17-s + (−0.170 + 0.295i)18-s + (−0.323 − 0.560i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.900 - 0.435i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.163431 + 0.712471i\)
\(L(\frac12)\) \(\approx\) \(0.163431 + 0.712471i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 + (0.132 + 0.229i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.45 - 2.51i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.717 - 1.24i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.75 - 4.76i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.41 - 4.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.41 + 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.99 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.04T + 29T^{2} \)
31 \( 1 + (4.60 - 7.97i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.306 + 0.530i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 + 8.43T + 43T^{2} \)
47 \( 1 + (1.20 + 2.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.914 + 1.58i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.435 + 0.754i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.66 - 2.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.31 + 5.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.85T + 71T^{2} \)
73 \( 1 + (1.57 - 2.72i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.78 - 15.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + (-0.497 - 0.861i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−10.67368721610650524040427815802, −10.28329980174148899576477878910, −9.698910226026898163280617700433, −8.720473748186767436066341546719, −7.03566171383749781126618250918, −6.38001245810156373846997535465, −5.25755305664208659623866457265, −4.80829074499015603334781662124, −3.41146337650272776619838796079, −1.98144921099396773280485559571, 0.42196071163705207331399610011, 2.00341341663886294148050333056, 3.17837679678233812712082468460, 4.99699894272297430956876460329, 5.94737776280281907394558279812, 6.62291071188226911183120661230, 7.49459175270699025889316682361, 8.283516617743011577983515618205, 8.912232684722707099112912743873, 10.57904239158398380145782526824

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