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Properties

Label 2-630-63.5-c1-0-13
Degree $2$
Conductor $630$
Sign $0.618 - 0.785i$
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  + i·2-s + (1.29 − 1.15i)3-s − 4-s + (0.5 + 0.866i)5-s + (1.15 + 1.29i)6-s + (1.14 + 2.38i)7-s − i·8-s + (0.338 − 2.98i)9-s + (−0.866 + 0.5i)10-s + (3.01 + 1.73i)11-s + (−1.29 + 1.15i)12-s + (−0.972 − 0.561i)13-s + (−2.38 + 1.14i)14-s + (1.64 + 0.542i)15-s + 16-s + (0.795 + 1.37i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.745 − 0.666i)3-s − 0.5·4-s + (0.223 + 0.387i)5-s + (0.470 + 0.527i)6-s + (0.431 + 0.902i)7-s − 0.353i·8-s + (0.112 − 0.993i)9-s + (−0.273 + 0.158i)10-s + (0.908 + 0.524i)11-s + (−0.372 + 0.333i)12-s + (−0.269 − 0.155i)13-s + (−0.638 + 0.304i)14-s + (0.424 + 0.139i)15-s + 0.250·16-s + (0.192 + 0.334i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.618 - 0.785i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.618 - 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80970 + 0.878789i\)
\(L(\frac12)\) \(\approx\) \(1.80970 + 0.878789i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.29 + 1.15i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-1.14 - 2.38i)T \)
good11 \( 1 + (-3.01 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.972 + 0.561i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.795 - 1.37i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.478 - 0.276i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.00422 - 0.00243i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.13 + 1.22i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.37iT - 31T^{2} \)
37 \( 1 + (-0.212 + 0.368i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.76 + 3.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.300 + 0.520i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + (-2.21 + 1.28i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.40T + 59T^{2} \)
61 \( 1 + 4.68iT - 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 + 14.9iT - 71T^{2} \)
73 \( 1 + (4.45 - 2.57i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 4.75T + 79T^{2} \)
83 \( 1 + (3.21 + 5.57i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.59 + 6.23i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.3 - 8.26i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.55188638061608187837161006577, −9.452007439907007788083681707285, −8.890403519505395710017902008778, −8.038590616102267813047464681727, −7.17863852017071308905208443779, −6.41501811073076957269780215369, −5.49337834744788605851560169976, −4.16953523733796613162149295829, −2.87475871989022446269094031493, −1.62568207716946860924382323382, 1.23854708579111552054003976099, 2.66956737396584400812281293381, 3.90912183368906505491457638242, 4.47241497000346720680140402400, 5.62628692075558323896677165240, 7.14773745749358790121139620108, 8.122983288751865801761060463452, 8.946457527833857067560016205236, 9.642158203680674045084409959543, 10.38822159487291559916256806934

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