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Properties

Label 2-630-105.2-c1-0-8
Degree $2$
Conductor $630$
Sign $0.993 - 0.116i$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−1.48 − 1.67i)5-s + (−2.30 + 1.30i)7-s + (−0.707 − 0.707i)8-s + (1.23 − 1.86i)10-s + (4.43 − 2.56i)11-s + (1.62 − 1.62i)13-s + (−1.85 − 1.88i)14-s + (0.500 − 0.866i)16-s + (6.99 + 1.87i)17-s + (−0.866 − 0.5i)19-s + (2.12 + 0.707i)20-s + (3.62 + 3.62i)22-s + (7.96 − 2.13i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.663 − 0.748i)5-s + (−0.870 + 0.492i)7-s + (−0.249 − 0.249i)8-s + (0.389 − 0.590i)10-s + (1.33 − 0.772i)11-s + (0.450 − 0.450i)13-s + (−0.495 − 0.504i)14-s + (0.125 − 0.216i)16-s + (1.69 + 0.454i)17-s + (−0.198 − 0.114i)19-s + (0.474 + 0.158i)20-s + (0.772 + 0.772i)22-s + (1.66 − 0.444i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.993 - 0.116i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.993 - 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35921 + 0.0792846i\)
\(L(\frac12)\) \(\approx\) \(1.35921 + 0.0792846i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 \)
5 \( 1 + (1.48 + 1.67i)T \)
7 \( 1 + (2.30 - 1.30i)T \)
good11 \( 1 + (-4.43 + 2.56i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.62 + 1.62i)T - 13iT^{2} \)
17 \( 1 + (-6.99 - 1.87i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.96 + 2.13i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.88 + 0.504i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 7.95iT - 41T^{2} \)
43 \( 1 + (-3 + 3i)T - 43iT^{2} \)
47 \( 1 + (2.32 + 8.69i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.65 - 9.89i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.00 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.62 - 8.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.56 - 9.56i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.41iT - 71T^{2} \)
73 \( 1 + (-3.06 - 0.821i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-13.2 - 7.62i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.29 - 2.29i)T + 83iT^{2} \)
89 \( 1 + (-4.94 + 8.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.24 - 3.24i)T + 97iT^{2} \)
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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

−10.60544583680191980723656280951, −9.265935917985230873831797888723, −8.932310346355112981220681793444, −8.004727513264965908547345687836, −7.02550348089599918797912642337, −6.00336215747663017460042414523, −5.34971610960091947144448328530, −3.93485642070611344191198838212, −3.29880909709286493837030023088, −0.890143929477253140555567076863, 1.26469263302394711485955631919, 3.11082692685220761543502229852, 3.65926007637260544878344707630, 4.72393580084922669211188961392, 6.23325415170865671431133426054, 6.96968077797273305136475448944, 7.82800646486514189769256499443, 9.301231423224898417056773600285, 9.657834907071981405317294281014, 10.73347553258718063932800140281

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