Properties

Label 2-630-105.104-c1-0-11
Degree $2$
Conductor $630$
Sign $-0.0250 + 0.999i$
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  − 2-s + 4-s − 2.23i·5-s + (1.58 − 2.12i)7-s − 8-s + 2.23i·10-s + 1.41i·11-s + 3.16·13-s + (−1.58 + 2.12i)14-s + 16-s − 4.47i·17-s − 2.23i·20-s − 1.41i·22-s − 6·23-s − 5.00·25-s − 3.16·26-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.999i·5-s + (0.597 − 0.801i)7-s − 0.353·8-s + 0.707i·10-s + 0.426i·11-s + 0.877·13-s + (−0.422 + 0.566i)14-s + 0.250·16-s − 1.08i·17-s − 0.499i·20-s − 0.301i·22-s − 1.25·23-s − 1.00·25-s − 0.620·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0250 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0250 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0250 + 0.999i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.0250 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.734716 - 0.753349i\)
\(L(\frac12)\) \(\approx\) \(0.734716 - 0.753349i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + 2.23iT \)
7 \( 1 + (-1.58 + 2.12i)T \)
good11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 - 3.16T + 13T^{2} \)
17 \( 1 + 4.47iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 4.24iT - 37T^{2} \)
41 \( 1 - 9.48T + 41T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 + 4.47iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 9.48T + 59T^{2} \)
61 \( 1 + 13.4iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + 6.32T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 8.94iT - 83T^{2} \)
89 \( 1 + 9.48T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−10.19019245128368008615632834623, −9.531236690328958957651902900364, −8.558511955016878264934298298333, −7.925117932707761620388852170211, −7.06977408034962942604364076625, −5.87469206114132527698504709593, −4.77625879851704816360809095817, −3.81808402610915940952594381156, −2.00033142574844190443925101690, −0.75552635560664395036088032976, 1.69990310217715932319361391860, 2.87145376950628508980656054161, 4.07690455696665579503348347992, 5.84316538225360915818631842009, 6.20168260108870078040702839289, 7.50703467499020019248506410240, 8.231401445990579585561071686273, 8.981022815354623021280794898161, 10.02241672483821613225587595146, 10.86960207532378637024513611208

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