Properties

Label 2-637-91.23-c1-0-28
Degree $2$
Conductor $637$
Sign $0.981 - 0.190i$
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  + 1.37i·2-s + (1.44 − 2.49i)3-s + 0.0982·4-s + (0.697 + 0.402i)5-s + (3.44 + 1.98i)6-s + 2.89i·8-s + (−2.65 − 4.59i)9-s + (−0.555 + 0.962i)10-s + (4.56 + 2.63i)11-s + (0.141 − 0.245i)12-s + (2.36 − 2.72i)13-s + (2.01 − 1.16i)15-s − 3.79·16-s − 0.560·17-s + (6.33 − 3.65i)18-s + (−5.06 + 2.92i)19-s + ⋯
L(s)  = 1  + 0.975i·2-s + (0.831 − 1.44i)3-s + 0.0491·4-s + (0.312 + 0.180i)5-s + (1.40 + 0.811i)6-s + 1.02i·8-s + (−0.883 − 1.53i)9-s + (−0.175 + 0.304i)10-s + (1.37 + 0.794i)11-s + (0.0408 − 0.0707i)12-s + (0.656 − 0.754i)13-s + (0.519 − 0.299i)15-s − 0.948·16-s − 0.135·17-s + (1.49 − 0.861i)18-s + (−1.16 + 0.670i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.981 - 0.190i$
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: \((2,\ 637,\ (\ :1/2),\ 0.981 - 0.190i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37101 + 0.228211i\)
\(L(\frac12)\) \(\approx\) \(2.37101 + 0.228211i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.36 + 2.72i)T \)
good2 \( 1 - 1.37iT - 2T^{2} \)
3 \( 1 + (-1.44 + 2.49i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.697 - 0.402i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.56 - 2.63i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.560T + 17T^{2} \)
19 \( 1 + (5.06 - 2.92i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.60T + 23T^{2} \)
29 \( 1 + (1.14 + 1.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.01 - 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.24iT - 37T^{2} \)
41 \( 1 + (0.803 - 0.463i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.22 + 3.85i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.32 - 1.92i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.72 + 4.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 10.9iT - 59T^{2} \)
61 \( 1 + (-3.65 - 6.32i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.36 - 3.67i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.06 + 4.65i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.33 - 2.50i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.68 - 9.84i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.81iT - 83T^{2} \)
89 \( 1 - 5.00iT - 89T^{2} \)
97 \( 1 + (9.22 + 5.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.60406161957942209884346493132, −9.313802900669141911756737295841, −8.451144548697679192699562337875, −7.905106656102725630072455366713, −6.89921166513320879113491126374, −6.53641496130286014861990761516, −5.67822150322306525044630277930, −3.94573913143547470395097870227, −2.47093829684847117510896947177, −1.57563599690564686817966741553, 1.63855227499729993726958633516, 2.92912954960312925912149100981, 3.83929419886280789209183367411, 4.38675962903769121020768745392, 5.92030268227841767267319328397, 6.94839886652046387558147655925, 8.560812850073515583803179991554, 9.148393261623028223429359062496, 9.577903779792248586410381910243, 10.71588472382860798238537180645

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