Properties

Label 2-637-7.4-c1-0-12
Degree $2$
Conductor $637$
Sign $-0.386 + 0.922i$
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  + (1.30 + 2.26i)2-s + (−1.11 + 1.93i)3-s + (−2.42 + 4.20i)4-s + (1.11 + 1.93i)5-s − 5.85·6-s − 7.47·8-s + (−1 − 1.73i)9-s + (−2.92 + 5.06i)10-s + (1.5 − 2.59i)11-s + (−5.42 − 9.39i)12-s + 13-s − 5.00·15-s + (−4.92 − 8.53i)16-s + (0.736 − 1.27i)17-s + (2.61 − 4.53i)18-s + (1.5 + 2.59i)19-s + ⋯
L(s)  = 1  + (0.925 + 1.60i)2-s + (−0.645 + 1.11i)3-s + (−1.21 + 2.10i)4-s + (0.499 + 0.866i)5-s − 2.38·6-s − 2.64·8-s + (−0.333 − 0.577i)9-s + (−0.925 + 1.60i)10-s + (0.452 − 0.783i)11-s + (−1.56 − 2.71i)12-s + 0.277·13-s − 1.29·15-s + (−1.23 − 2.13i)16-s + (0.178 − 0.309i)17-s + (0.617 − 1.06i)18-s + (0.344 + 0.596i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.386 + 0.922i$
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: \((2,\ 637,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06804 - 1.60564i\)
\(L(\frac12)\) \(\approx\) \(1.06804 - 1.60564i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 - 2.26i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.11 - 1.93i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.11 - 1.93i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.736 + 1.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.11 - 7.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.35 + 4.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (3.73 + 6.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.73 + 6.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.736 - 1.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 + (1.35 - 2.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.35 - 2.34i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-1.11 - 1.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.41T + 97T^{2} \)
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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

−11.24787954536451516340043757789, −10.22058679508675968019203746393, −9.388029193672217040499140666298, −8.367781993969385686579002588395, −7.29659759714342890931595266705, −6.43656087010451697196656727330, −5.71177127466275601571271082886, −5.09806230723001197364171513873, −3.95607249836668106297901538249, −3.20905148485737465340353365424, 0.949291099823486747223309816262, 1.65101066267164557495071182343, 2.91831266741466623778743797806, 4.45700126473678639780881347558, 5.07530185526766253034894590276, 6.11586339675428518223589975536, 6.93076587463328469304648565271, 8.547058384835467774897959403444, 9.415776598239538169082884989601, 10.28591927399010808511086711188

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