Properties

Label 2-637-91.80-c1-0-36
Degree $2$
Conductor $637$
Sign $-0.560 + 0.828i$
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  + (0.0302 + 0.112i)2-s − 2.59i·3-s + (1.72 − 0.993i)4-s + (−0.456 + 1.70i)5-s + (0.293 − 0.0785i)6-s + (0.329 + 0.329i)8-s − 3.75·9-s − 0.206·10-s + (−1.38 − 1.38i)11-s + (−2.58 − 4.47i)12-s + (−1.85 − 3.09i)13-s + (4.43 + 1.18i)15-s + (1.95 − 3.39i)16-s + (−2.13 − 3.70i)17-s + (−0.113 − 0.423i)18-s + (−3.01 − 3.01i)19-s + ⋯
L(s)  = 1  + (0.0213 + 0.0797i)2-s − 1.50i·3-s + (0.860 − 0.496i)4-s + (−0.204 + 0.762i)5-s + (0.119 − 0.0320i)6-s + (0.116 + 0.116i)8-s − 1.25·9-s − 0.0651·10-s + (−0.417 − 0.417i)11-s + (−0.745 − 1.29i)12-s + (−0.515 − 0.857i)13-s + (1.14 + 0.306i)15-s + (0.489 − 0.848i)16-s + (−0.518 − 0.898i)17-s + (−0.0267 − 0.0997i)18-s + (−0.692 − 0.692i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.560 + 0.828i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.560 + 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.722645 - 1.36195i\)
\(L(\frac12)\) \(\approx\) \(0.722645 - 1.36195i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.85 + 3.09i)T \)
good2 \( 1 + (-0.0302 - 0.112i)T + (-1.73 + i)T^{2} \)
3 \( 1 + 2.59iT - 3T^{2} \)
5 \( 1 + (0.456 - 1.70i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.38 + 1.38i)T + 11iT^{2} \)
17 \( 1 + (2.13 + 3.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.01 + 3.01i)T + 19iT^{2} \)
23 \( 1 + (-5.53 - 3.19i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.57 + 6.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.13 + 1.10i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.73 + 0.732i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.94 - 11.0i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.55 + 0.896i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.40 - 1.71i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.13 - 3.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.62 - 0.436i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 3.08iT - 61T^{2} \)
67 \( 1 + (-0.0139 + 0.0139i)T - 67iT^{2} \)
71 \( 1 + (-1.23 - 4.59i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.255 - 0.954i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.96 - 5.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.87 - 9.87i)T + 83iT^{2} \)
89 \( 1 + (-2.07 - 7.76i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-14.2 + 3.82i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.57832997048370042800558860555, −9.449277207242805847884435687751, −8.066306456832740898898309746779, −7.44182903909339585880461555720, −6.80442028322732090964957294853, −6.12777406929629938732572442850, −5.05700484191984333494282166130, −2.92998350861227052652622646074, −2.39281494052670671454290836499, −0.808205715459855066019805482903, 2.08287957997819394116303310017, 3.46084178134154080763356990719, 4.36623005586500875117876063436, 5.05045060821894937363609194488, 6.37234128615554375923044495093, 7.39590077075866876008542352773, 8.623609265276863513863660283978, 9.002044172836194933133277619285, 10.33851056345582411608736427801, 10.60320496999738686603859350959

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