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Properties

Label 2-630-105.23-c1-0-13
Degree $2$
Conductor $630$
Sign $0.610 + 0.792i$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (1.01 − 1.99i)5-s + (2.58 − 0.576i)7-s + (0.707 − 0.707i)8-s + (0.469 − 2.18i)10-s + (−1.41 + 0.819i)11-s + (1 + i)13-s + (2.34 − 1.22i)14-s + (0.500 − 0.866i)16-s + (−0.599 + 2.23i)17-s + (−0.274 − 0.158i)19-s + (−0.111 − 2.23i)20-s + (−1.15 + 1.15i)22-s + (−2.15 − 8.03i)23-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.433 − 0.249i)4-s + (0.456 − 0.889i)5-s + (0.976 − 0.217i)7-s + (0.249 − 0.249i)8-s + (0.148 − 0.691i)10-s + (−0.427 + 0.246i)11-s + (0.277 + 0.277i)13-s + (0.626 − 0.327i)14-s + (0.125 − 0.216i)16-s + (−0.145 + 0.542i)17-s + (−0.0629 − 0.0363i)19-s + (−0.0250 − 0.499i)20-s + (−0.246 + 0.246i)22-s + (−0.448 − 1.67i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.610 + 0.792i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.610 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29472 - 1.12857i\)
\(L(\frac12)\) \(\approx\) \(2.29472 - 1.12857i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 \)
5 \( 1 + (-1.01 + 1.99i)T \)
7 \( 1 + (-2.58 + 0.576i)T \)
good11 \( 1 + (1.41 - 0.819i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + (0.599 - 2.23i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.274 + 0.158i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.15 + 8.03i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 1.19T + 29T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.83 - 6.83i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.82iT - 41T^{2} \)
43 \( 1 + (-5.63 - 5.63i)T + 43iT^{2} \)
47 \( 1 + (-6.10 + 1.63i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (11.2 + 3.01i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.04 - 1.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.158 + 0.274i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.59 - 0.963i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.51iT - 71T^{2} \)
73 \( 1 + (3.41 - 12.7i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.34 + 4.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.0 - 10.0i)T - 83iT^{2} \)
89 \( 1 + (7.74 - 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.15 - 9.15i)T - 97iT^{2} \)
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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

−10.59550529699837450129710844721, −9.753040258608655437207571904003, −8.571109147851395190564645070929, −8.001869300267005768218487758317, −6.68399858447868677908770933347, −5.74107618688116003675633281045, −4.74780499963722745371987161345, −4.21768266957564393589787700439, −2.48715142615147256742866955936, −1.33705099849698325564488505438, 1.91367978868100126529304274770, 3.00682440322579936456714423795, 4.17700624673890474811677092375, 5.47433936394455807086155332546, 5.90835651911394257784425480087, 7.25998519926867261483044130738, 7.75147740149831658196861510094, 8.956435892488063674325181752860, 10.01084103041552741389697562862, 11.03524395987225276406072458596

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