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Properties

Label 2-630-35.3-c1-0-0
Degree $2$
Conductor $630$
Sign $-0.588 - 0.808i$
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 + (−1.55 − 1.61i)5-s + (−1.38 − 2.25i)7-s + (−0.707 − 0.707i)8-s + (1.08 + 1.95i)10-s + (−0.582 + 1.00i)11-s + (−1.92 + 1.92i)13-s + (0.756 + 2.53i)14-s + (0.500 + 0.866i)16-s + (0.0209 − 0.00560i)17-s + (−0.989 − 1.71i)19-s + (−0.537 − 2.17i)20-s + (0.824 − 0.824i)22-s + (−1.85 + 6.93i)23-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.433 + 0.249i)4-s + (−0.693 − 0.720i)5-s + (−0.524 − 0.851i)7-s + (−0.249 − 0.249i)8-s + (0.341 + 0.618i)10-s + (−0.175 + 0.304i)11-s + (−0.533 + 0.533i)13-s + (0.202 + 0.677i)14-s + (0.125 + 0.216i)16-s + (0.00507 − 0.00136i)17-s + (−0.226 − 0.393i)19-s + (−0.120 − 0.485i)20-s + (0.175 − 0.175i)22-s + (−0.387 + 1.44i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.588 - 0.808i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.588 - 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0492886 + 0.0968803i\)
\(L(\frac12)\) \(\approx\) \(0.0492886 + 0.0968803i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.55 + 1.61i)T \)
7 \( 1 + (1.38 + 2.25i)T \)
good11 \( 1 + (0.582 - 1.00i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.92 - 1.92i)T - 13iT^{2} \)
17 \( 1 + (-0.0209 + 0.00560i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.989 + 1.71i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.85 - 6.93i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.60iT - 29T^{2} \)
31 \( 1 + (-6.86 - 3.96i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (10.2 + 2.74i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.48iT - 41T^{2} \)
43 \( 1 + (7.87 + 7.87i)T + 43iT^{2} \)
47 \( 1 + (-1.05 + 3.94i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.82 - 0.757i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.34 - 9.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.15 - 1.82i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.76 - 14.0i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.51T + 71T^{2} \)
73 \( 1 + (-0.969 - 3.61i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.39 - 0.805i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.74 + 9.74i)T - 83iT^{2} \)
89 \( 1 + (1.80 + 3.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.265 + 0.265i)T + 97iT^{2} \)
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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

−10.71753067694333510592097168214, −10.03048965889454513891511367384, −9.137277089427227676943784908989, −8.410898712718138643522341834823, −7.30031209310952116764047894993, −6.94008423568542680072720115353, −5.34195340615892936160603267633, −4.24639431768772147880287850241, −3.25410934018056334359886298003, −1.52230859448220617796066258713, 0.07313478492625684787321701612, 2.38635067397268110461533825881, 3.27936277031046373324698930893, 4.77698230992713317797413492199, 6.15898214984803281457174185910, 6.62711728884489414273849943085, 7.997116654647158417917064147000, 8.241295945468987728272880581172, 9.520997693676629827848814187113, 10.20979719635761237841301513524

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