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Properties

Label 2-630-315.194-c1-0-8
Degree $2$
Conductor $630$
Sign $-0.0906 - 0.995i$
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 + (−1.13 + 1.30i)3-s + 4-s + (−2.11 + 0.716i)5-s + (1.13 − 1.30i)6-s + (2.41 + 1.07i)7-s − 8-s + (−0.410 − 2.97i)9-s + (2.11 − 0.716i)10-s + (5.13 + 2.96i)11-s + (−1.13 + 1.30i)12-s + (2.57 − 4.46i)13-s + (−2.41 − 1.07i)14-s + (1.47 − 3.58i)15-s + 16-s + (−1.62 + 0.935i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.656 + 0.753i)3-s + 0.5·4-s + (−0.947 + 0.320i)5-s + (0.464 − 0.533i)6-s + (0.912 + 0.408i)7-s − 0.353·8-s + (−0.136 − 0.990i)9-s + (0.669 − 0.226i)10-s + (1.54 + 0.894i)11-s + (−0.328 + 0.376i)12-s + (0.715 − 1.23i)13-s + (−0.645 − 0.288i)14-s + (0.380 − 0.924i)15-s + 0.250·16-s + (−0.392 + 0.226i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0906 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0906 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0906 - 0.995i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.0906 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.544713 + 0.596543i\)
\(L(\frac12)\) \(\approx\) \(0.544713 + 0.596543i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.13 - 1.30i)T \)
5 \( 1 + (2.11 - 0.716i)T \)
7 \( 1 + (-2.41 - 1.07i)T \)
good11 \( 1 + (-5.13 - 2.96i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.57 + 4.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.62 - 0.935i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.51 + 0.872i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.20 - 5.55i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.16 - 1.25i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.36iT - 31T^{2} \)
37 \( 1 + (-1.17 - 0.680i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.79 - 6.57i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.07 - 2.35i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.74iT - 47T^{2} \)
53 \( 1 + (0.628 + 1.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.30T + 59T^{2} \)
61 \( 1 - 11.4iT - 61T^{2} \)
67 \( 1 - 15.0iT - 67T^{2} \)
71 \( 1 - 2.88iT - 71T^{2} \)
73 \( 1 + (1.79 + 3.11i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 8.42T + 79T^{2} \)
83 \( 1 + (2.43 - 1.40i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.92 + 15.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.23 + 2.13i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.84925025393665469591222390228, −10.08287557373467081822152431131, −9.004751096393542196738089083629, −8.435739294239330953189924106179, −7.30793400032923885469959445916, −6.48826668679110180381027493361, −5.31982412255096873664197966328, −4.26530722033563646345853110923, −3.29313593028396560553772123327, −1.30406161590029145709373090117, 0.73160805755363768870820319631, 1.79652837976618901087852440831, 3.81917984877954669422653488796, 4.74992722568712894167619862631, 6.26133990439050627513118457754, 6.81894335940714185066717104188, 7.79694972482861018021215895584, 8.567501559730503113924925807335, 9.161381009174282793480761120775, 10.87036732038880910094184549564

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