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Properties

Label 2-637-91.30-c1-0-35
Degree $2$
Conductor $637$
Sign $-0.794 + 0.606i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.89 + 1.09i)2-s + 1.79·3-s + (1.39 − 2.41i)4-s + (−1.89 − 1.09i)5-s + (−3.39 + 1.96i)6-s + 1.73i·8-s + 0.208·9-s + 4.79·10-s + 1.27i·11-s + (2.5 − 4.33i)12-s + (−3.5 + 0.866i)13-s + (−3.39 − 1.96i)15-s + (0.895 + 1.55i)16-s + (1.5 − 2.59i)17-s + (−0.395 + 0.228i)18-s + 6.56i·19-s + ⋯
L(s)  = 1  + (−1.34 + 0.773i)2-s + 1.03·3-s + (0.697 − 1.20i)4-s + (−0.847 − 0.489i)5-s + (−1.38 + 0.800i)6-s + 0.612i·8-s + 0.0695·9-s + 1.51·10-s + 0.384i·11-s + (0.721 − 1.24i)12-s + (−0.970 + 0.240i)13-s + (−0.876 − 0.506i)15-s + (0.223 + 0.387i)16-s + (0.363 − 0.630i)17-s + (−0.0932 + 0.0538i)18-s + 1.50i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 + 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.794 + 0.606i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.794 + 0.606i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 - 0.866i)T \)
good2 \( 1 + (1.89 - 1.09i)T + (1 - 1.73i)T^{2} \)
3 \( 1 - 1.79T + 3T^{2} \)
5 \( 1 + (1.89 + 1.09i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 1.27iT - 11T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 6.56iT - 19T^{2} \)
23 \( 1 + (3.79 + 6.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.10 + 1.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.5 - 4.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6 - 3.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.20 + 1.27i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.18 - 3.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.70 + 2.14i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.08 - 10.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.66 + 4.42i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 12.7T + 61T^{2} \)
67 \( 1 + 11.4iT - 67T^{2} \)
71 \( 1 + (0.791 - 0.456i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3 + 1.73i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.55iT - 83T^{2} \)
89 \( 1 + (2.52 - 1.45i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.1 + 7.61i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.849549336082513644982882104581, −9.123845962071346004289905342567, −8.385808931351345474648786529357, −7.82587392553078210127500271384, −7.25343191510540273514513274793, −6.04682563206607060888512777965, −4.62374025012526072344057157228, −3.43820993912466889096361815679, −1.90064063071080855486757651577, 0, 1.96800491764636992644598015215, 3.00238638961305347992632021906, 3.72838439381942006519464339291, 5.45459392373462712278001370540, 7.25024355995263600117105028241, 7.64677250007074815033225172550, 8.487364802641742903168912823373, 9.188918583482049933505662723934, 9.874856597244581503091418665687

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