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Properties

Label 2-630-63.5-c1-0-11
Degree $2$
Conductor $630$
Sign $0.980 + 0.195i$
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 + (−1.14 + 1.30i)3-s − 4-s + (0.5 + 0.866i)5-s + (1.30 + 1.14i)6-s + (2.06 − 1.64i)7-s + i·8-s + (−0.390 − 2.97i)9-s + (0.866 − 0.5i)10-s + (0.568 + 0.328i)11-s + (1.14 − 1.30i)12-s + (−2.39 − 1.38i)13-s + (−1.64 − 2.06i)14-s + (−1.69 − 0.338i)15-s + 16-s + (3.36 + 5.83i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.659 + 0.751i)3-s − 0.5·4-s + (0.223 + 0.387i)5-s + (0.531 + 0.466i)6-s + (0.782 − 0.622i)7-s + 0.353i·8-s + (−0.130 − 0.991i)9-s + (0.273 − 0.158i)10-s + (0.171 + 0.0989i)11-s + (0.329 − 0.375i)12-s + (−0.664 − 0.383i)13-s + (−0.440 − 0.553i)14-s + (−0.438 − 0.0873i)15-s + 0.250·16-s + (0.817 + 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.980 + 0.195i$
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: \((2,\ 630,\ (\ :1/2),\ 0.980 + 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27276 - 0.125430i\)
\(L(\frac12)\) \(\approx\) \(1.27276 - 0.125430i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.14 - 1.30i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.06 + 1.64i)T \)
good11 \( 1 + (-0.568 - 0.328i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.39 + 1.38i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.36 - 5.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.633 + 0.365i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.23 + 3.60i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.21 + 3.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.01iT - 31T^{2} \)
37 \( 1 + (-1.05 + 1.81i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.35 + 5.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.29 - 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.85T + 47T^{2} \)
53 \( 1 + (-5.77 + 3.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.25T + 59T^{2} \)
61 \( 1 - 1.56iT - 61T^{2} \)
67 \( 1 + 6.56T + 67T^{2} \)
71 \( 1 - 1.54iT - 71T^{2} \)
73 \( 1 + (-5.95 + 3.43i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 + (-8.71 - 15.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.32 + 2.29i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.02 - 2.32i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.59445464052978182042160883616, −10.12624299965415392383934112405, −9.083711660784403558636027169824, −8.099645905364283551148093177250, −6.93580274356299227230867022801, −5.81016451328204880823986872294, −4.82401740261262051394238571932, −4.05366591343850193403043977606, −2.85372192410978793576714017939, −1.11893466849447484371606246172, 1.05537196282345838924784132394, 2.58311134757073394502720557457, 4.63293400589554920963903711881, 5.26656606849711705966226787037, 6.03120346623924962208014859478, 7.20372682099517303218106326399, 7.69179346130174314562582673523, 8.795101560281209509681326619268, 9.497878519563125442453112839666, 10.72488106872361399091222935655

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