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Properties

Label 2-630-315.223-c1-0-10
Degree $2$
Conductor $630$
Sign $0.709 - 0.704i$
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 + (0.684 + 1.59i)3-s + (−0.866 + 0.499i)4-s + (−0.438 − 2.19i)5-s + (1.35 − 1.07i)6-s + (−2.53 − 0.745i)7-s + (0.707 + 0.707i)8-s + (−2.06 + 2.17i)9-s + (−2.00 + 0.991i)10-s + (−2.30 + 3.99i)11-s + (−1.38 − 1.03i)12-s + (4.25 + 1.14i)13-s + (−0.0626 + 2.64i)14-s + (3.18 − 2.19i)15-s + (0.500 − 0.866i)16-s + (5.57 + 5.57i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.395 + 0.918i)3-s + (−0.433 + 0.249i)4-s + (−0.196 − 0.980i)5-s + (0.554 − 0.438i)6-s + (−0.959 − 0.281i)7-s + (0.249 + 0.249i)8-s + (−0.687 + 0.726i)9-s + (−0.633 + 0.313i)10-s + (−0.695 + 1.20i)11-s + (−0.400 − 0.298i)12-s + (1.18 + 0.316i)13-s + (−0.0167 + 0.706i)14-s + (0.823 − 0.567i)15-s + (0.125 − 0.216i)16-s + (1.35 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.709 - 0.704i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.709 - 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07268 + 0.442393i\)
\(L(\frac12)\) \(\approx\) \(1.07268 + 0.442393i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.684 - 1.59i)T \)
5 \( 1 + (0.438 + 2.19i)T \)
7 \( 1 + (2.53 + 0.745i)T \)
good11 \( 1 + (2.30 - 3.99i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.25 - 1.14i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-5.57 - 5.57i)T + 17iT^{2} \)
19 \( 1 - 7.15T + 19T^{2} \)
23 \( 1 + (1.29 - 4.82i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.14 - 1.24i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.31 + 0.761i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.17 + 3.17i)T - 37iT^{2} \)
41 \( 1 + (0.829 - 0.478i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.54 - 1.48i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (8.74 - 2.34i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.426 + 0.426i)T + 53iT^{2} \)
59 \( 1 + (2.62 + 4.54i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.46 - 2.57i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.46 + 2.53i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.91T + 71T^{2} \)
73 \( 1 + (1.32 - 1.32i)T - 73iT^{2} \)
79 \( 1 + (-12.2 - 7.09i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.96 + 7.34i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 3.99T + 89T^{2} \)
97 \( 1 + (-5.19 + 1.39i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.40196273599217999004813302966, −9.790572264316233999572837525447, −9.334744925186477678714437734232, −8.243048591804332633095328735441, −7.61256283335805496013734211053, −5.86664818944107518649970855448, −4.94960328522963564534855072830, −3.86724553457244627350750194424, −3.25055042120871213609606908433, −1.49373251880455258563002324800, 0.70352198994029998764063908079, 3.03154914969577925189668724992, 3.25848402257083808564667757282, 5.50608864445843172869899092733, 6.16333563419565958680173261046, 6.94289843871691332178946000660, 7.81810214481520630813906210919, 8.421350116338877752003639237225, 9.517286577401735757238683631670, 10.30628599201680462615677298511

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