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Properties

Label 2-630-5.4-c1-0-6
Degree $2$
Conductor $630$
Sign $0.894 + 0.447i$
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 − 4-s + (2 + i)5-s + i·7-s + i·8-s + (1 − 2i)10-s − 4i·13-s + 14-s + 16-s + 2i·17-s + 8·19-s + (−2 − i)20-s + 8i·23-s + (3 + 4i)25-s − 4·26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.894 + 0.447i)5-s + 0.377i·7-s + 0.353i·8-s + (0.316 − 0.632i)10-s − 1.10i·13-s + 0.267·14-s + 0.250·16-s + 0.485i·17-s + 1.83·19-s + (−0.447 − 0.223i)20-s + 1.66i·23-s + (0.600 + 0.800i)25-s − 0.784·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64819 - 0.389086i\)
\(L(\frac12)\) \(\approx\) \(1.64819 - 0.389086i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-2 - i)T \)
7 \( 1 - iT \)
good11 \( 1 + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−10.31529745828327732583015865858, −9.941061038410829275019050461368, −9.036282714935392713844527018644, −8.029479390995284509235287394639, −6.97825568682782207222386409766, −5.67725156428613387770483250389, −5.22957916698797422259489754979, −3.52409716305951997408299658200, −2.72103708481317336112328359056, −1.35996701240880706455183564253, 1.19424583573088466237925119486, 2.89000593295966485594978736565, 4.52203207450589931304947774670, 5.09090846982921197241254871485, 6.37258544847786355920627883817, 6.85270210861731788899936460411, 8.092437036545450606450883741632, 8.871692668389388938023742599383, 9.748709981893755748827259663325, 10.28407581988392245336394221870

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