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Properties

Label 2-630-63.25-c1-0-28
Degree $2$
Conductor $630$
Sign $0.580 + 0.814i$
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 + (1.5 − 0.866i)3-s + 4-s + (0.5 − 0.866i)5-s + (1.5 − 0.866i)6-s + (−2 − 1.73i)7-s + 8-s + (1.5 − 2.59i)9-s + (0.5 − 0.866i)10-s + (1.5 − 0.866i)12-s + (2 + 3.46i)13-s + (−2 − 1.73i)14-s − 1.73i·15-s + 16-s + (1.5 − 2.59i)18-s + (−1 − 1.73i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.866 − 0.499i)3-s + 0.5·4-s + (0.223 − 0.387i)5-s + (0.612 − 0.353i)6-s + (−0.755 − 0.654i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s + (0.158 − 0.273i)10-s + (0.433 − 0.249i)12-s + (0.554 + 0.960i)13-s + (−0.534 − 0.462i)14-s − 0.447i·15-s + 0.250·16-s + (0.353 − 0.612i)18-s + (−0.229 − 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.580 + 0.814i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.580 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60243 - 1.34104i\)
\(L(\frac12)\) \(\approx\) \(2.60243 - 1.34104i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2 + 1.73i)T \)
good11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.46190338462439932277847286279, −9.445033729965284520195419349202, −8.791522890318179597822811015662, −7.67287270462553408894963336702, −6.77327113186497511185373468528, −6.19246350730544000035282143921, −4.66090084177411322117201729768, −3.77702980781685845132090211119, −2.74891981447445444763778048110, −1.36481776270053802479604340386, 2.15580621080420397833725850882, 3.15936419588161073453390506441, 3.86387976294663709543627960013, 5.26629747983766552686987831856, 6.03093633686515806676927030567, 7.14190411991154944198345905385, 8.140950325875924112646368212599, 9.024300868298668252337431464618, 9.975280292588606817397177220846, 10.55356176675862854782324236799

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