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Properties

Label 2-637-91.23-c1-0-14
Degree $2$
Conductor $637$
Sign $0.372 - 0.927i$
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  + 2.18i·2-s + (0.895 − 1.55i)3-s − 2.79·4-s + (−1.89 − 1.09i)5-s + (3.39 + 1.96i)6-s − 1.73i·8-s + (−0.104 − 0.180i)9-s + (2.39 − 4.14i)10-s + (1.10 + 0.637i)11-s + (−2.49 + 4.33i)12-s + (3.5 + 0.866i)13-s + (−3.39 + 1.96i)15-s − 1.79·16-s + 3·17-s + (0.395 − 0.228i)18-s + (5.68 − 3.28i)19-s + ⋯
L(s)  = 1  + 1.54i·2-s + (0.517 − 0.895i)3-s − 1.39·4-s + (−0.847 − 0.489i)5-s + (1.38 + 0.800i)6-s − 0.612i·8-s + (−0.0347 − 0.0602i)9-s + (0.757 − 1.31i)10-s + (0.332 + 0.192i)11-s + (−0.721 + 1.24i)12-s + (0.970 + 0.240i)13-s + (−0.876 + 0.506i)15-s − 0.447·16-s + 0.727·17-s + (0.0932 − 0.0538i)18-s + (1.30 − 0.753i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.372 - 0.927i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.372 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33406 + 0.901622i\)
\(L(\frac12)\) \(\approx\) \(1.33406 + 0.901622i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.18iT - 2T^{2} \)
3 \( 1 + (-0.895 + 1.55i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.89 + 1.09i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.10 - 0.637i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + (-5.68 + 3.28i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.58T + 23T^{2} \)
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 \( 1 - 6.92iT - 37T^{2} \)
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 \( 1 - 8.85iT - 59T^{2} \)
61 \( 1 + (6.37 + 11.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.87 + 5.70i)T + (33.5 + 58.0i)T^{2} \)
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 \( 1 + 2.91iT - 89T^{2} \)
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

−10.79526417684735664833950029497, −9.159331419569469170663478863689, −8.735078549943441784494869676900, −7.81721471721968327102241989906, −7.31145815459036313551312016574, −6.64517179320774365373490598418, −5.42908137548725399376775896928, −4.56578541770108484399372854762, −3.19851392988863822957567174270, −1.22207996422282042602728488688, 1.14960984070049545214224612595, 2.97442661593573578698350622305, 3.56650979106340368876028914644, 4.10882791166648357287218098820, 5.49956221424824451192111860178, 7.04585765583933580504961077853, 8.105569118275456025999076142019, 9.192701300471768681622905416372, 9.616816874499505150984702890608, 10.61076657448359154929199690277

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