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Properties

Label 2-637-49.43-c1-0-24
Degree $2$
Conductor $637$
Sign $0.164 + 0.986i$
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  + (0.153 − 0.192i)2-s + (−1.67 − 0.804i)3-s + (0.431 + 1.89i)4-s + (0.217 + 0.104i)5-s + (−0.410 + 0.197i)6-s + (−2.59 − 0.499i)7-s + (0.873 + 0.420i)8-s + (0.271 + 0.340i)9-s + (0.0536 − 0.0258i)10-s + (−0.00113 + 0.00142i)11-s + (0.799 − 3.50i)12-s + (0.623 − 0.781i)13-s + (−0.494 + 0.423i)14-s + (−0.279 − 0.350i)15-s + (−3.27 + 1.57i)16-s + (1.38 − 6.06i)17-s + ⋯
L(s)  = 1  + (0.108 − 0.136i)2-s + (−0.964 − 0.464i)3-s + (0.215 + 0.945i)4-s + (0.0974 + 0.0469i)5-s + (−0.167 + 0.0807i)6-s + (−0.981 − 0.188i)7-s + (0.308 + 0.148i)8-s + (0.0905 + 0.113i)9-s + (0.0169 − 0.00816i)10-s + (−0.000342 + 0.000429i)11-s + (0.230 − 1.01i)12-s + (0.172 − 0.216i)13-s + (−0.132 + 0.113i)14-s + (−0.0721 − 0.0904i)15-s + (−0.819 + 0.394i)16-s + (0.335 − 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $0.164 + 0.986i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ 0.164 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.668847 - 0.566732i\)
\(L(\frac12)\) \(\approx\) \(0.668847 - 0.566732i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.59 + 0.499i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
good2 \( 1 + (-0.153 + 0.192i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (1.67 + 0.804i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (-0.217 - 0.104i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (0.00113 - 0.00142i)T + (-2.44 - 10.7i)T^{2} \)
17 \( 1 + (-1.38 + 6.06i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 7.36T + 19T^{2} \)
23 \( 1 + (1.18 + 5.20i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (0.125 - 0.550i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 9.65T + 31T^{2} \)
37 \( 1 + (-1.63 + 7.16i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (9.56 + 4.60i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (3.15 - 1.52i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-5.99 + 7.51i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (0.930 + 4.07i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (2.09 - 1.00i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-0.251 + 1.10i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + 5.22T + 67T^{2} \)
71 \( 1 + (-0.583 - 2.55i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (4.06 + 5.10i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 - 0.670T + 79T^{2} \)
83 \( 1 + (-7.51 - 9.42i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (9.78 + 12.2i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 12.5T + 97T^{2} \)
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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

−10.47683390964656515542621586036, −9.656736075502128726651599433795, −8.588527379544852194440776575343, −7.44143528145403242407521868382, −6.85676960157564038235480317675, −6.00287376679626588850562355861, −4.92958312817787969039010385651, −3.55695210286730476572641695430, −2.66433509083248418962364609805, −0.56010057257215692930628145526, 1.36256692260714262802518441333, 3.18139684256286275337368533596, 4.52993211345875248607639432231, 5.62385198058605073882077868202, 5.94902664969694814934186300989, 6.87155335216403172423919455168, 8.113973407402111580454029347622, 9.556520821283198102653413449311, 9.900531845833704683107022079296, 10.66777027438668112110522761237

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