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Properties

Label 2-645-43.6-c1-0-15
Degree $2$
Conductor $645$
Sign $-0.127 + 0.991i$
Analytic cond. $5.15035$
Root an. cond. $2.26943$
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  − 0.802·2-s + (0.5 − 0.866i)3-s − 1.35·4-s + (−0.5 + 0.866i)5-s + (−0.401 + 0.694i)6-s + (−0.385 − 0.666i)7-s + 2.69·8-s + (−0.499 − 0.866i)9-s + (0.401 − 0.694i)10-s + 1.61·11-s + (−0.678 + 1.17i)12-s + (0.940 + 1.62i)13-s + (0.308 + 0.534i)14-s + (0.499 + 0.866i)15-s + 0.554·16-s + (−2.63 − 4.56i)17-s + ⋯
L(s)  = 1  − 0.567·2-s + (0.288 − 0.499i)3-s − 0.678·4-s + (−0.223 + 0.387i)5-s + (−0.163 + 0.283i)6-s + (−0.145 − 0.252i)7-s + 0.951·8-s + (−0.166 − 0.288i)9-s + (0.126 − 0.219i)10-s + 0.485·11-s + (−0.195 + 0.339i)12-s + (0.260 + 0.451i)13-s + (0.0825 + 0.142i)14-s + (0.129 + 0.223i)15-s + 0.138·16-s + (−0.638 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(645\)    =    \(3 \cdot 5 \cdot 43\)
Sign: $-0.127 + 0.991i$
Analytic conductor: \(5.15035\)
Root analytic conductor: \(2.26943\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{645} (436, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 645,\ (\ :1/2),\ -0.127 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.512020 - 0.582267i\)
\(L(\frac12)\) \(\approx\) \(0.512020 - 0.582267i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-2.75 + 5.95i)T \)
good2 \( 1 + 0.802T + 2T^{2} \)
7 \( 1 + (0.385 + 0.666i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.61T + 11T^{2} \)
13 \( 1 + (-0.940 - 1.62i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.63 + 4.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.208 - 0.360i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.54 + 2.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.77 + 6.54i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.580 + 1.00i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.93 + 10.2i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.51T + 41T^{2} \)
47 \( 1 + 2.66T + 47T^{2} \)
53 \( 1 + (0.00924 - 0.0160i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.15T + 59T^{2} \)
61 \( 1 + (-3.53 - 6.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.50 - 4.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.67 + 11.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.05 - 10.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.53 + 6.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.69 + 8.13i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.800 + 1.38i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−10.16858411015173818647386829168, −9.250988761059295797813525831813, −8.767792463541356837602483098800, −7.65722447659419643391305211215, −7.08960656648238330691866300659, −6.00214432143742750763417413208, −4.58299310397996530363275718540, −3.70645409727147627650960868579, −2.20095628833446753250843133508, −0.54771878407180008309902684554, 1.42663591279856744151763591184, 3.29617651251507358497705644090, 4.27914921685874226635416065413, 5.13515549091036474633645451063, 6.31236970786891526100970990226, 7.66271692894300852355313402712, 8.438078353143814127928629517719, 9.037227006785956412133827206590, 9.722909586305291073471846607313, 10.61913629221958931990557886165

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