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Properties

Label 2-630-7.3-c2-0-25
Degree $2$
Conductor $630$
Sign $-0.999 - 0.0230i$
Analytic cond. $17.1662$
Root an. cond. $4.14321$
Motivic weight $2$
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 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (1.93 − 1.11i)5-s + (−6.51 + 2.55i)7-s + 2.82·8-s + (−2.73 − 1.58i)10-s + (6.16 − 10.6i)11-s + 7.26i·13-s + (7.73 + 6.17i)14-s + (−2.00 − 3.46i)16-s + (−8.04 − 4.64i)17-s + (5.26 − 3.03i)19-s + 4.47i·20-s − 17.4·22-s + (−1.12 − 1.94i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.930 + 0.365i)7-s + 0.353·8-s + (−0.273 − 0.158i)10-s + (0.560 − 0.970i)11-s + 0.558i·13-s + (0.552 + 0.440i)14-s + (−0.125 − 0.216i)16-s + (−0.473 − 0.273i)17-s + (0.276 − 0.159i)19-s + 0.223i·20-s − 0.792·22-s + (−0.0489 − 0.0847i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0230i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.999 - 0.0230i$
Analytic conductor: \(17.1662\)
Root analytic conductor: \(4.14321\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1),\ -0.999 - 0.0230i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5019626792\)
\(L(\frac12)\) \(\approx\) \(0.5019626792\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
3 \( 1 \)
5 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (6.51 - 2.55i)T \)
good11 \( 1 + (-6.16 + 10.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 7.26iT - 169T^{2} \)
17 \( 1 + (8.04 + 4.64i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-5.26 + 3.03i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (1.12 + 1.94i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 42.2T + 841T^{2} \)
31 \( 1 + (1.05 + 0.609i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (17.5 + 30.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 57.8iT - 1.68e3T^{2} \)
43 \( 1 + 34.0T + 1.84e3T^{2} \)
47 \( 1 + (49.4 - 28.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-7.27 + 12.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (50.1 + 28.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (5.07 - 2.93i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (24.7 - 42.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 101.T + 5.04e3T^{2} \)
73 \( 1 + (-71.2 - 41.1i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (55.8 + 96.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 91.6iT - 6.88e3T^{2} \)
89 \( 1 + (110. - 63.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 61.4iT - 9.40e3T^{2} \)
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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

−9.808377520259056706661327932532, −9.140960662225784413879698723241, −8.648948851654192338304541556645, −7.28972000871223364544863975311, −6.34011680281308772998385040686, −5.41215287081478207923301143767, −4.00771310845367409344065922247, −3.05992101995921515205985393654, −1.80086940559862466425125420329, −0.20336021582573748816525337982, 1.61050367448338281964114430888, 3.20513070416504467756176661030, 4.41214172026144661286215952721, 5.61154084297669586632304709164, 6.53961642031055060782634392598, 7.13365221994649978903807017758, 8.123486477467815920255011266108, 9.275213260296846371503386030403, 9.800497316239573389437162317305, 10.49756999256280119679164696316

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