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Properties

Label 2-637-91.88-c1-0-20
Degree $2$
Conductor $637$
Sign $-0.824 - 0.566i$
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  + (−1.19 − 0.689i)2-s − 2.88·3-s + (−0.0491 − 0.0850i)4-s + (−0.697 + 0.402i)5-s + (3.44 + 1.98i)6-s + 2.89i·8-s + 5.30·9-s + 1.11·10-s − 5.27i·11-s + (0.141 + 0.245i)12-s + (2.36 − 2.72i)13-s + (2.01 − 1.16i)15-s + (1.89 − 3.28i)16-s + (0.280 + 0.485i)17-s + (−6.33 − 3.65i)18-s − 5.84i·19-s + ⋯
L(s)  = 1  + (−0.844 − 0.487i)2-s − 1.66·3-s + (−0.0245 − 0.0425i)4-s + (−0.312 + 0.180i)5-s + (1.40 + 0.811i)6-s + 1.02i·8-s + 1.76·9-s + 0.351·10-s − 1.58i·11-s + (0.0408 + 0.0707i)12-s + (0.656 − 0.754i)13-s + (0.519 − 0.299i)15-s + (0.474 − 0.821i)16-s + (0.0679 + 0.117i)17-s + (−1.49 − 0.861i)18-s − 1.34i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.824 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.824 - 0.566i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.824 - 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0447328 + 0.144152i\)
\(L(\frac12)\) \(\approx\) \(0.0447328 + 0.144152i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 + 0.689i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + 2.88T + 3T^{2} \)
5 \( 1 + (0.697 - 0.402i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 5.27iT - 11T^{2} \)
17 \( 1 + (-0.280 - 0.485i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 5.84iT - 19T^{2} \)
23 \( 1 + (0.802 - 1.38i)T + (-11.5 - 19.9i)T^{2} \)
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 \( 1 + (-1.07 - 0.620i)T + (18.5 + 32.0i)T^{2} \)
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 \( 1 + (9.52 - 5.49i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 7.30T + 61T^{2} \)
67 \( 1 + 7.34iT - 67T^{2} \)
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 \( 1 + (4.33 + 2.50i)T + (44.5 + 77.0i)T^{2} \)
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

−10.43817994843934870094093200815, −9.363532218998560009259016612041, −8.471398001335411889194880443041, −7.48238386518554884044393973081, −6.16011569608929446637325300262, −5.69788138497467010612299407991, −4.69896195564806640700607450577, −3.15668939565791682070067643918, −1.17185208351360676201559859086, −0.16941092966093580448331921964, 1.48468076077077028196617772411, 4.01997017550761137142769346771, 4.67288849804720277198202453107, 5.99201110899995052070227356753, 6.68333481826606675631954504535, 7.48425045696469546802622495532, 8.345248230783472386001733238499, 9.599729592795075241932026326028, 10.06508672011972468859616939398, 11.00204777782873654032664962962

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