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Properties

Label 2-637-91.74-c1-0-15
Degree $2$
Conductor $637$
Sign $-0.446 + 0.894i$
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  − 2.38·2-s + (−1.37 − 2.38i)3-s + 3.70·4-s + (0.491 + 0.850i)5-s + (3.28 + 5.69i)6-s − 4.06·8-s + (−2.28 + 3.95i)9-s + (−1.17 − 2.03i)10-s + (0.293 + 0.509i)11-s + (−5.09 − 8.82i)12-s + (−2.39 − 2.69i)13-s + (1.35 − 2.34i)15-s + 2.30·16-s + 6.45·17-s + (5.45 − 9.45i)18-s + (−1.91 + 3.31i)19-s + ⋯
L(s)  = 1  − 1.68·2-s + (−0.794 − 1.37i)3-s + 1.85·4-s + (0.219 + 0.380i)5-s + (1.34 + 2.32i)6-s − 1.43·8-s + (−0.761 + 1.31i)9-s + (−0.370 − 0.642i)10-s + (0.0886 + 0.153i)11-s + (−1.47 − 2.54i)12-s + (−0.663 − 0.748i)13-s + (0.348 − 0.604i)15-s + 0.576·16-s + 1.56·17-s + (1.28 − 2.22i)18-s + (−0.438 + 0.760i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.446 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.446 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.446 + 0.894i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.446 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.236339 - 0.382203i\)
\(L(\frac12)\) \(\approx\) \(0.236339 - 0.382203i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.39 + 2.69i)T \)
good2 \( 1 + 2.38T + 2T^{2} \)
3 \( 1 + (1.37 + 2.38i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.491 - 0.850i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.293 - 0.509i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 6.45T + 17T^{2} \)
19 \( 1 + (1.91 - 3.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.26T + 23T^{2} \)
29 \( 1 + (-1.98 + 3.42i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.49 - 2.58i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.75T + 37T^{2} \)
41 \( 1 + (-1.83 + 3.17i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.19 + 5.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.17 + 3.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.212 - 0.368i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.00T + 59T^{2} \)
61 \( 1 + (-1.10 + 1.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.50 + 6.07i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.80 + 3.11i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.46 + 4.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.39 + 2.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.86T + 83T^{2} \)
89 \( 1 - 2.09T + 89T^{2} \)
97 \( 1 + (-3.84 - 6.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.39018230015637958657774647299, −9.485694219361260173286374479438, −8.313650180022205195460836120606, −7.66417064301226906472507306355, −7.02185861002321662716328443441, −6.26992335167983122880050226015, −5.26826412248891716297775080852, −2.87563965479997819645837348140, −1.66812778624682861465806820869, −0.57832258904084789940288647068, 1.13233025484344767286346958763, 3.00386571948342543341490619828, 4.56893948519993417608427761921, 5.37511774993004170745678955478, 6.57375014664706111164782828600, 7.50444130050188588347225510034, 8.691109192798054706003010870064, 9.404901114621574191550601537788, 9.744741104598919442297633200152, 10.70088526121553460678397398932

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