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Properties

Label 2-637-7.4-c1-0-0
Degree $2$
Conductor $637$
Sign $0.605 + 0.795i$
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  + (1.32 + 2.29i)2-s + (−1.19 + 2.07i)3-s + (−2.52 + 4.37i)4-s + (−1.82 − 3.16i)5-s − 6.36·6-s − 8.10·8-s + (−1.37 − 2.37i)9-s + (4.85 − 8.40i)10-s + (−0.327 + 0.567i)11-s + (−6.05 − 10.4i)12-s − 13-s + 8.75·15-s + (−5.70 − 9.88i)16-s + (−1.19 + 2.07i)17-s + (3.63 − 6.30i)18-s + (1.35 + 2.34i)19-s + ⋯
L(s)  = 1  + (0.938 + 1.62i)2-s + (−0.691 + 1.19i)3-s + (−1.26 + 2.18i)4-s + (−0.817 − 1.41i)5-s − 2.59·6-s − 2.86·8-s + (−0.456 − 0.791i)9-s + (1.53 − 2.65i)10-s + (−0.0988 + 0.171i)11-s + (−1.74 − 3.02i)12-s − 0.277·13-s + 2.26·15-s + (−1.42 − 2.47i)16-s + (−0.290 + 0.503i)17-s + (0.857 − 1.48i)18-s + (0.310 + 0.537i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.605 + 0.795i$
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: \((2,\ 637,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.408789 - 0.202647i\)
\(L(\frac12)\) \(\approx\) \(0.408789 - 0.202647i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 - 2.29i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.19 - 2.07i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.82 + 3.16i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.327 - 0.567i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.19 - 2.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.35 - 2.34i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.68 + 6.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.208T + 29T^{2} \)
31 \( 1 + (-0.568 + 0.984i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.72 - 6.44i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 3.10T + 43T^{2} \)
47 \( 1 + (2.30 + 3.98i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.62 - 4.55i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.12 - 7.15i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.948 - 1.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.44 + 11.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.75T + 71T^{2} \)
73 \( 1 + (6.26 - 10.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.759 - 1.31i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.7T + 83T^{2} \)
89 \( 1 + (-7.40 - 12.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−11.72892229826304670095027978906, −10.32793182589722236471608852695, −9.298603896222744139248339022057, −8.378965389340972895748740422468, −7.902172242616274661162761795685, −6.59877199414082053929289835024, −5.63950365973091143030525576815, −4.84600175098474077328786385672, −4.43055169186965200662577830710, −3.64314792404633435687460281059, 0.19524746124879706762616610652, 1.82461341601746379129452685230, 2.92005616738070379685673315941, 3.76513866867843848473978182481, 5.07571938015031445044820164257, 6.10945373788534304965244059756, 6.94997380370416874544885279001, 7.81455459710515191297329728054, 9.440285385431193561552392757425, 10.37205698153630817217161212774

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