Properties

Label 2-637-91.83-c1-0-35
Degree $2$
Conductor $637$
Sign $-0.00675 + 0.999i$
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  + (1.90 − 1.90i)2-s + 0.759i·3-s − 5.26i·4-s + (1.78 + 1.78i)5-s + (1.44 + 1.44i)6-s + (−6.22 − 6.22i)8-s + 2.42·9-s + 6.81·10-s + (1.52 + 1.52i)11-s + 3.99·12-s + (−1.44 − 3.30i)13-s + (−1.35 + 1.35i)15-s − 13.1·16-s − 1.40·17-s + (4.61 − 4.61i)18-s + (1.48 + 1.48i)19-s + ⋯
L(s)  = 1  + (1.34 − 1.34i)2-s + 0.438i·3-s − 2.63i·4-s + (0.799 + 0.799i)5-s + (0.590 + 0.590i)6-s + (−2.19 − 2.19i)8-s + 0.807·9-s + 2.15·10-s + (0.459 + 0.459i)11-s + 1.15·12-s + (−0.401 − 0.915i)13-s + (−0.350 + 0.350i)15-s − 3.29·16-s − 0.339·17-s + (1.08 − 1.08i)18-s + (0.339 + 0.339i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00675 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00675 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.00675 + 0.999i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.00675 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.39397 - 2.41019i\)
\(L(\frac12)\) \(\approx\) \(2.39397 - 2.41019i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.44 + 3.30i)T \)
good2 \( 1 + (-1.90 + 1.90i)T - 2iT^{2} \)
3 \( 1 - 0.759iT - 3T^{2} \)
5 \( 1 + (-1.78 - 1.78i)T + 5iT^{2} \)
11 \( 1 + (-1.52 - 1.52i)T + 11iT^{2} \)
17 \( 1 + 1.40T + 17T^{2} \)
19 \( 1 + (-1.48 - 1.48i)T + 19iT^{2} \)
23 \( 1 - 1.31iT - 23T^{2} \)
29 \( 1 + 4.56T + 29T^{2} \)
31 \( 1 + (5.14 + 5.14i)T + 31iT^{2} \)
37 \( 1 + (-1.61 - 1.61i)T + 37iT^{2} \)
41 \( 1 + (-2.69 - 2.69i)T + 41iT^{2} \)
43 \( 1 + 0.437iT - 43T^{2} \)
47 \( 1 + (5.66 - 5.66i)T - 47iT^{2} \)
53 \( 1 + 2.53T + 53T^{2} \)
59 \( 1 + (5.52 - 5.52i)T - 59iT^{2} \)
61 \( 1 - 7.58iT - 61T^{2} \)
67 \( 1 + (-0.401 + 0.401i)T - 67iT^{2} \)
71 \( 1 + (10.7 - 10.7i)T - 71iT^{2} \)
73 \( 1 + (-8.70 + 8.70i)T - 73iT^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + (-3.82 - 3.82i)T + 83iT^{2} \)
89 \( 1 + (-0.0366 + 0.0366i)T - 89iT^{2} \)
97 \( 1 + (9.43 + 9.43i)T + 97iT^{2} \)
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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

−10.57012417757960687405786166431, −9.759325992122443516196489246523, −9.472315742877569769384643565121, −7.41670826415207183730073095414, −6.31872129588308687045664768014, −5.52791802346257551342462764592, −4.54896874545013537363776144903, −3.64938558321898916815212722981, −2.63617492307830397567148318426, −1.58798955291024804401632733291, 1.94142731709443839556008772563, 3.61239673811507537355579639927, 4.63278850405630326098019371527, 5.33014077907097575723150649674, 6.34017474887762118308703324612, 6.93957835135333602717198913993, 7.79017028679083103685624214659, 8.886105752426764874354195875673, 9.477837015217707472516742048790, 11.16150857612454657110135760790

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