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Properties

Label 2-630-15.2-c1-0-0
Degree $2$
Conductor $630$
Sign $-0.812 - 0.583i$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.489 + 2.18i)5-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−1.19 − 1.88i)10-s + 1.77i·11-s + (0.692 − 0.692i)13-s − 1.00·14-s − 1.00·16-s + (−2.39 + 2.39i)17-s + (2.18 + 0.489i)20-s + (−1.25 − 1.25i)22-s + (3.38 + 3.38i)23-s + (−4.52 − 2.13i)25-s + 0.979i·26-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.218 + 0.975i)5-s + (0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + (−0.378 − 0.597i)10-s + 0.536i·11-s + (0.192 − 0.192i)13-s − 0.267·14-s − 0.250·16-s + (−0.580 + 0.580i)17-s + (0.487 + 0.109i)20-s + (−0.268 − 0.268i)22-s + (0.705 + 0.705i)23-s + (−0.904 − 0.427i)25-s + 0.192i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.812 - 0.583i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.812 - 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262181 + 0.815036i\)
\(L(\frac12)\) \(\approx\) \(0.262181 + 0.815036i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (0.489 - 2.18i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good11 \( 1 - 1.77iT - 11T^{2} \)
13 \( 1 + (-0.692 + 0.692i)T - 13iT^{2} \)
17 \( 1 + (2.39 - 2.39i)T - 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-3.38 - 3.38i)T + 23iT^{2} \)
29 \( 1 + 4.42T + 29T^{2} \)
31 \( 1 + 5.77T + 31T^{2} \)
37 \( 1 + (-5.91 - 5.91i)T + 37iT^{2} \)
41 \( 1 + 0.807iT - 41T^{2} \)
43 \( 1 + (4.64 - 4.64i)T - 43iT^{2} \)
47 \( 1 + (7.47 - 7.47i)T - 47iT^{2} \)
53 \( 1 + (-3.56 - 3.56i)T + 53iT^{2} \)
59 \( 1 + 5.89T + 59T^{2} \)
61 \( 1 - 2.39T + 61T^{2} \)
67 \( 1 + (0.641 + 0.641i)T + 67iT^{2} \)
71 \( 1 + 10.6iT - 71T^{2} \)
73 \( 1 + (-5.70 + 5.70i)T - 73iT^{2} \)
79 \( 1 - 16.3iT - 79T^{2} \)
83 \( 1 + (0.171 + 0.171i)T + 83iT^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + (-12.6 - 12.6i)T + 97iT^{2} \)
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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

−11.01538064817413724308524909318, −9.970479559243995950305655934228, −9.239533241827665447597737469924, −8.164501478314267613442535243521, −7.45202980309130647200560796094, −6.61362698810677415303000260066, −5.75230375805997776637615989852, −4.53989150674039207607873383671, −3.21787415023205365356389318800, −1.82587699350995261144045061136, 0.55125469474807721398218536233, 1.98008798669324644269097612238, 3.52200809705384642020716782197, 4.52867362730415129420484539348, 5.52422621544513497250946652246, 6.89325695505496737222287237594, 7.82609543864487874748800971604, 8.737842236243217879060876142437, 9.188275216769363662900857832783, 10.25269936499797651936922668023

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