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Properties

Label 2-630-63.5-c1-0-17
Degree $2$
Conductor $630$
Sign $0.280 + 0.959i$
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  + i·2-s + (−1.69 + 0.353i)3-s − 4-s + (0.5 + 0.866i)5-s + (−0.353 − 1.69i)6-s + (−1.70 + 2.02i)7-s − i·8-s + (2.74 − 1.19i)9-s + (−0.866 + 0.5i)10-s + (−4.48 − 2.58i)11-s + (1.69 − 0.353i)12-s + (−0.604 − 0.349i)13-s + (−2.02 − 1.70i)14-s + (−1.15 − 1.29i)15-s + 16-s + (1.77 + 3.07i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.978 + 0.204i)3-s − 0.5·4-s + (0.223 + 0.387i)5-s + (−0.144 − 0.692i)6-s + (−0.643 + 0.765i)7-s − 0.353i·8-s + (0.916 − 0.399i)9-s + (−0.273 + 0.158i)10-s + (−1.35 − 0.780i)11-s + (0.489 − 0.102i)12-s + (−0.167 − 0.0968i)13-s + (−0.541 − 0.454i)14-s + (−0.298 − 0.333i)15-s + 0.250·16-s + (0.430 + 0.745i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.280 + 0.959i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.280 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.172056 - 0.129016i\)
\(L(\frac12)\) \(\approx\) \(0.172056 - 0.129016i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (1.69 - 0.353i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.70 - 2.02i)T \)
good11 \( 1 + (4.48 + 2.58i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.604 + 0.349i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.77 - 3.07i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.54 + 2.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.98 + 2.87i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.60 - 2.65i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.55iT - 31T^{2} \)
37 \( 1 + (-5.06 + 8.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.18 + 5.52i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.94 + 10.3i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.16T + 47T^{2} \)
53 \( 1 + (8.82 - 5.09i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.35T + 59T^{2} \)
61 \( 1 - 6.10iT - 61T^{2} \)
67 \( 1 + 4.42T + 67T^{2} \)
71 \( 1 + 4.76iT - 71T^{2} \)
73 \( 1 + (3.19 - 1.84i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + (-7.80 - 13.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.57 - 4.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (15.5 - 9.00i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.68406715039397713585272923243, −9.509430459766825806876260725405, −8.717205676182267841211852360693, −7.56963215225723688258940104319, −6.65510739074056055759341694256, −5.75022562417973695314226162847, −5.39424526763408096171216561873, −3.99991548007760342995054159752, −2.59537742784178307394640444642, −0.13587071515667176080424360546, 1.42161380097206082956165890308, 2.93768983643690390740488957916, 4.44368361128479836846747029623, 5.06720174244453790239140137188, 6.17984189143038698420872566125, 7.24611995520979032945554100596, 8.022813754309900831826021151610, 9.561565470310212824607856507930, 9.997887717101384153882212609080, 10.76185237617227557604650066913

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