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Properties

Label 2-630-15.8-c1-0-0
Degree $2$
Conductor $630$
Sign $-0.939 - 0.343i$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−1.52 + 1.63i)5-s + (−0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + (2.23 − 0.0743i)10-s − 2.14i·11-s + (2.16 + 2.16i)13-s + 1.00·14-s − 1.00·16-s + (−4.46 − 4.46i)17-s + (−1.63 − 1.52i)20-s + (−1.51 + 1.51i)22-s + (−6.32 + 6.32i)23-s + (−0.332 − 4.98i)25-s − 3.05i·26-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.683 + 0.730i)5-s + (−0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + (0.706 − 0.0234i)10-s − 0.647i·11-s + (0.599 + 0.599i)13-s + 0.267·14-s − 0.250·16-s + (−1.08 − 1.08i)17-s + (−0.365 − 0.341i)20-s + (−0.323 + 0.323i)22-s + (−1.31 + 1.31i)23-s + (−0.0664 − 0.997i)25-s − 0.599i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.939 - 0.343i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.939 - 0.343i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0182779 + 0.103250i\)
\(L(\frac12)\) \(\approx\) \(0.0182779 + 0.103250i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 + (1.52 - 1.63i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 + 2.14iT - 11T^{2} \)
13 \( 1 + (-2.16 - 2.16i)T + 13iT^{2} \)
17 \( 1 + (4.46 + 4.46i)T + 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (6.32 - 6.32i)T - 23iT^{2} \)
29 \( 1 + 8.20T + 29T^{2} \)
31 \( 1 + 1.85T + 31T^{2} \)
37 \( 1 + (5.13 - 5.13i)T - 37iT^{2} \)
41 \( 1 + 7.56iT - 41T^{2} \)
43 \( 1 + (7.84 + 7.84i)T + 43iT^{2} \)
47 \( 1 + (-5.01 - 5.01i)T + 47iT^{2} \)
53 \( 1 + (2.35 - 2.35i)T - 53iT^{2} \)
59 \( 1 - 9.35T + 59T^{2} \)
61 \( 1 + 4.46T + 61T^{2} \)
67 \( 1 + (3.84 - 3.84i)T - 67iT^{2} \)
71 \( 1 + 0.420iT - 71T^{2} \)
73 \( 1 + (2.63 + 2.63i)T + 73iT^{2} \)
79 \( 1 - 5.23iT - 79T^{2} \)
83 \( 1 + (10.6 - 10.6i)T - 83iT^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 + (0.606 - 0.606i)T - 97iT^{2} \)
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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

−11.16728128222009007998370949337, −10.17600067559217227356167494462, −9.223542178299573308263334305922, −8.535903095803622243428972828292, −7.47971894687797442188618826961, −6.77127597128031827534349736887, −5.62477584778743548757132941542, −4.06009362316699420348584035194, −3.31303399147512238075945902519, −2.03946750636815436410520698697, 0.06472119865622048867312195910, 1.81473785056282991519299659493, 3.75174089535357437951249842306, 4.59451088486242280964172496819, 5.80227904323510286204019803707, 6.71504038614500802713990230646, 7.74462420816331209248056161948, 8.397915490587192278999324606612, 9.138260898467966609155486373323, 10.17139777041768121973031991050

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