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Properties

Label 2-637-13.3-c1-0-23
Degree $2$
Conductor $637$
Sign $0.727 - 0.686i$
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  + (0.289 + 0.502i)2-s + (0.946 + 1.63i)3-s + (0.831 − 1.44i)4-s + 1.47·5-s + (−0.548 + 0.950i)6-s + 2.12·8-s + (−0.289 + 0.502i)9-s + (0.427 + 0.739i)10-s + (−0.289 − 0.502i)11-s + 3.14·12-s + (−0.128 + 3.60i)13-s + (1.39 + 2.41i)15-s + (−1.04 − 1.81i)16-s + (0.598 − 1.03i)17-s − 0.336·18-s + (0.230 − 0.399i)19-s + ⋯
L(s)  = 1  + (0.204 + 0.355i)2-s + (0.546 + 0.946i)3-s + (0.415 − 0.720i)4-s + 0.659·5-s + (−0.223 + 0.387i)6-s + 0.751·8-s + (−0.0966 + 0.167i)9-s + (0.135 + 0.233i)10-s + (−0.0874 − 0.151i)11-s + 0.908·12-s + (−0.0357 + 0.999i)13-s + (0.359 + 0.623i)15-s + (−0.261 − 0.453i)16-s + (0.145 − 0.251i)17-s − 0.0792·18-s + (0.0528 − 0.0915i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.727 - 0.686i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.727 - 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31562 + 0.919598i\)
\(L(\frac12)\) \(\approx\) \(2.31562 + 0.919598i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.128 - 3.60i)T \)
good2 \( 1 + (-0.289 - 0.502i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.946 - 1.63i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 1.47T + 5T^{2} \)
11 \( 1 + (0.289 + 0.502i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.598 + 1.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.230 + 0.399i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.18 + 2.05i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.44 - 5.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.44T + 31T^{2} \)
37 \( 1 + (4.58 + 7.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.00 - 3.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.02 - 6.97i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + 9.39T + 53T^{2} \)
59 \( 1 + (0.120 - 0.208i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.86 - 6.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.724 - 1.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.25 + 10.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.69T + 73T^{2} \)
79 \( 1 - 16.0T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + (1.24 + 2.15i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.82 - 13.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.48269374612427527758620041950, −9.724018601394590073266889043170, −9.273182458155870853294894801469, −8.161876495958905586269346762800, −6.87152506967442415187739369277, −6.21804703500295942248998175848, −5.09382296465203305565283742638, −4.34880512575019808267548990958, −2.98727383168121233517606937412, −1.66158221927596276142838311504, 1.59058216955735166432475461521, 2.50572230772502125393385968961, 3.45227645664299585687775169474, 4.88202757789733824037848100683, 6.17255330241092875729253294727, 7.01649251790707416892268119677, 8.048024213274473977572146526860, 8.248361771669931834203090836789, 9.782022238773617990460042799171, 10.43483234266453222337866063878

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