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Properties

Label 2-637-91.20-c1-0-7
Degree $2$
Conductor $637$
Sign $0.312 - 0.949i$
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  + (1.94 − 0.521i)2-s + (−1.25 − 0.721i)3-s + (1.78 − 1.03i)4-s + (−2.32 + 2.32i)5-s + (−2.81 − 0.753i)6-s + (0.0872 − 0.0872i)8-s + (−0.457 − 0.793i)9-s + (−3.30 + 5.72i)10-s + (1.53 + 5.74i)11-s − 2.97·12-s + (2.81 + 2.25i)13-s + (4.57 − 1.22i)15-s + (−1.93 + 3.35i)16-s + (0.314 + 0.544i)17-s + (−1.30 − 1.30i)18-s + (0.712 + 0.191i)19-s + ⋯
L(s)  = 1  + (1.37 − 0.368i)2-s + (−0.721 − 0.416i)3-s + (0.892 − 0.515i)4-s + (−1.03 + 1.03i)5-s + (−1.14 − 0.307i)6-s + (0.0308 − 0.0308i)8-s + (−0.152 − 0.264i)9-s + (−1.04 + 1.81i)10-s + (0.463 + 1.73i)11-s − 0.859·12-s + (0.779 + 0.626i)13-s + (1.18 − 0.316i)15-s + (−0.484 + 0.838i)16-s + (0.0762 + 0.132i)17-s + (−0.307 − 0.307i)18-s + (0.163 + 0.0438i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.312 - 0.949i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.312 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24534 + 0.901328i\)
\(L(\frac12)\) \(\approx\) \(1.24534 + 0.901328i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.81 - 2.25i)T \)
good2 \( 1 + (-1.94 + 0.521i)T + (1.73 - i)T^{2} \)
3 \( 1 + (1.25 + 0.721i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.32 - 2.32i)T - 5iT^{2} \)
11 \( 1 + (-1.53 - 5.74i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.314 - 0.544i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.712 - 0.191i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.93 + 2.27i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.33 - 2.31i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.781 - 0.781i)T - 31iT^{2} \)
37 \( 1 + (-0.655 - 2.44i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.746 + 2.78i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.49 + 2.01i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.20 + 5.20i)T + 47iT^{2} \)
53 \( 1 + 7.78T + 53T^{2} \)
59 \( 1 + (-0.246 + 0.919i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.943 + 0.544i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.21 + 1.39i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.590 + 2.20i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.35 - 4.35i)T + 73iT^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + (-3.59 + 3.59i)T - 83iT^{2} \)
89 \( 1 + (-3.94 + 1.05i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-8.41 - 2.25i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.22966889937556992404678801809, −10.37234943353080376350401909665, −9.080867682416849845116006657504, −7.74305062743769862981474632936, −6.71083015204568553176105104937, −6.40594475023648021615739709874, −5.09279631088894822538211230448, −4.07243766842167118456343117754, −3.43916619026974758668395077233, −1.98969127417750471168930430451, 0.58757076205605080050977240691, 3.28624258783383315599903249835, 4.01495150608894136023853053185, 4.89258420714187126246530521593, 5.72278344367006143726896080108, 6.21639901652613488004943187314, 7.79754280121532436824853852835, 8.414436972841218651520056591517, 9.494990372458985039158833944953, 10.97675543273642834992379989491

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