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Properties

Label 2-630-315.212-c1-0-36
Degree $2$
Conductor $630$
Sign $-0.432 + 0.901i$
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  + (0.707 − 0.707i)2-s + (−0.895 + 1.48i)3-s − 1.00i·4-s + (2.15 − 0.610i)5-s + (0.415 + 1.68i)6-s + (−2.60 − 0.441i)7-s + (−0.707 − 0.707i)8-s + (−1.39 − 2.65i)9-s + (1.08 − 1.95i)10-s + (−5.45 − 3.14i)11-s + (1.48 + 0.895i)12-s + (−1.53 − 5.72i)13-s + (−2.15 + 1.53i)14-s + (−1.02 + 3.73i)15-s − 1.00·16-s + (4.19 + 1.12i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.516 + 0.856i)3-s − 0.500i·4-s + (0.962 − 0.272i)5-s + (0.169 + 0.686i)6-s + (−0.985 − 0.166i)7-s + (−0.250 − 0.250i)8-s + (−0.465 − 0.884i)9-s + (0.344 − 0.617i)10-s + (−1.64 − 0.949i)11-s + (0.428 + 0.258i)12-s + (−0.425 − 1.58i)13-s + (−0.576 + 0.409i)14-s + (−0.263 + 0.964i)15-s − 0.250·16-s + (1.01 + 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.432 + 0.901i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.432 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.619284 - 0.984287i\)
\(L(\frac12)\) \(\approx\) \(0.619284 - 0.984287i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.895 - 1.48i)T \)
5 \( 1 + (-2.15 + 0.610i)T \)
7 \( 1 + (2.60 + 0.441i)T \)
good11 \( 1 + (5.45 + 3.14i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.53 + 5.72i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-4.19 - 1.12i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.36 - 0.787i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.303 + 1.13i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.461 - 0.800i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.20T + 31T^{2} \)
37 \( 1 + (-2.33 + 0.625i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.94 + 3.43i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.81 - 0.485i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-4.13 + 4.13i)T - 47iT^{2} \)
53 \( 1 + (1.58 - 5.91i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + 8.32T + 59T^{2} \)
61 \( 1 + 12.3T + 61T^{2} \)
67 \( 1 + (-4.51 - 4.51i)T + 67iT^{2} \)
71 \( 1 + 9.47iT - 71T^{2} \)
73 \( 1 + (-5.78 - 1.54i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 - 13.3iT - 79T^{2} \)
83 \( 1 + (-4.66 + 17.4i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-6.67 + 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.0223 + 0.0834i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.45768935178670874684603526353, −9.872903568704759874449979387922, −8.920252414350763368260520675212, −7.70070677287101744180431343613, −6.10297627292805491568156544476, −5.60692774658591175362293622820, −5.00212747728499024579360472875, −3.39104142038606711190289706564, −2.82399925326283033466280430744, −0.53995886973478949991103454870, 2.02993689623524881735006330488, 2.95840498690140069673773929572, 4.81893409200155545375475513602, 5.53746829564719851202610364843, 6.43016042169879842406032253918, 7.11287453307435526349650281157, 7.80635790371118691255210955977, 9.271150139335199841662158003796, 9.933257350498109376335198445976, 10.91804919518022559965138723416

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