Properties

Label 2-630-63.58-c1-0-19
Degree $2$
Conductor $630$
Sign $0.0218 + 0.999i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
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  − 2-s + (0.478 + 1.66i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.478 − 1.66i)6-s + (−2.56 − 0.658i)7-s − 8-s + (−2.54 + 1.59i)9-s + (0.5 + 0.866i)10-s + (1.11 − 1.93i)11-s + (0.478 + 1.66i)12-s + (0.263 − 0.456i)13-s + (2.56 + 0.658i)14-s + (1.20 − 1.24i)15-s + 16-s + (−2.56 − 4.44i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.275 + 0.961i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.195 − 0.679i)6-s + (−0.968 − 0.249i)7-s − 0.353·8-s + (−0.847 + 0.530i)9-s + (0.158 + 0.273i)10-s + (0.336 − 0.583i)11-s + (0.137 + 0.480i)12-s + (0.0730 − 0.126i)13-s + (0.684 + 0.176i)14-s + (0.310 − 0.321i)15-s + 0.250·16-s + (−0.621 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0218 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0218 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.0218 + 0.999i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.0218 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.370670 - 0.362667i\)
\(L(\frac12)\) \(\approx\) \(0.370670 - 0.362667i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.478 - 1.66i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.56 + 0.658i)T \)
good11 \( 1 + (-1.11 + 1.93i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.263 + 0.456i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.56 + 4.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.263 + 0.456i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.82 + 6.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.08 - 1.87i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.275T + 31T^{2} \)
37 \( 1 + (1.07 - 1.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.55 + 9.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.51 - 2.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.971T + 47T^{2} \)
53 \( 1 + (5.80 + 10.0i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 9.37T + 59T^{2} \)
61 \( 1 - 2.10T + 61T^{2} \)
67 \( 1 - 2.64T + 67T^{2} \)
71 \( 1 + 0.00533T + 71T^{2} \)
73 \( 1 + (2.19 + 3.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + (-7.55 - 13.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.23 + 12.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.85 + 4.95i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.25828439834151323991243008879, −9.407852298088939560239495485477, −8.879826809179602543229425770443, −8.049449628651490276902939693977, −6.87617565819794384168353003856, −5.92924854098899143601392530684, −4.68139159897792839240394154043, −3.62161672504840049082330596732, −2.59027166148629034227906074434, −0.33296095607326357615030538926, 1.62850325287363171828606148187, 2.79528406747245815343377405370, 3.92169751214699775071029945134, 5.93027872855666779256201335620, 6.47004107905224605031930313494, 7.37463926654908535406884513184, 8.082616207629900243543792321512, 9.109156871323277001064275083528, 9.717282494770504811711664234047, 10.77987066620113393224912931838

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