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Properties

Label 2-630-35.27-c1-0-6
Degree $2$
Conductor $630$
Sign $-0.390 - 0.920i$
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 + (1.28 + 1.83i)5-s + (0.0564 + 2.64i)7-s + (−0.707 + 0.707i)8-s + (−0.386 + 2.20i)10-s + 1.41·11-s + (−0.772 − 0.772i)13-s + (−1.83 + 1.91i)14-s − 1.00·16-s + (−0.546 + 0.546i)17-s − 5.95·19-s + (−1.83 + 1.28i)20-s + (1.00 + 1.00i)22-s + (5.23 − 5.23i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.574 + 0.818i)5-s + (0.0213 + 0.999i)7-s + (−0.250 + 0.250i)8-s + (−0.122 + 0.696i)10-s + 0.426·11-s + (−0.214 − 0.214i)13-s + (−0.489 + 0.510i)14-s − 0.250·16-s + (−0.132 + 0.132i)17-s − 1.36·19-s + (−0.409 + 0.287i)20-s + (0.213 + 0.213i)22-s + (1.09 − 1.09i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.390 - 0.920i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.390 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10853 + 1.67453i\)
\(L(\frac12)\) \(\approx\) \(1.10853 + 1.67453i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.28 - 1.83i)T \)
7 \( 1 + (-0.0564 - 2.64i)T \)
good11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 + (0.772 + 0.772i)T + 13iT^{2} \)
17 \( 1 + (0.546 - 0.546i)T - 17iT^{2} \)
19 \( 1 + 5.95T + 19T^{2} \)
23 \( 1 + (-5.23 + 5.23i)T - 23iT^{2} \)
29 \( 1 - 0.992iT - 29T^{2} \)
31 \( 1 - 2.85iT - 31T^{2} \)
37 \( 1 + (-5.70 - 5.70i)T + 37iT^{2} \)
41 \( 1 - 2.18iT - 41T^{2} \)
43 \( 1 + (2 - 2i)T - 43iT^{2} \)
47 \( 1 + (-7.32 + 7.32i)T - 47iT^{2} \)
53 \( 1 + (-2.40 + 2.40i)T - 53iT^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 + 3.63iT - 61T^{2} \)
67 \( 1 + (4 + 4i)T + 67iT^{2} \)
71 \( 1 + 8.06T + 71T^{2} \)
73 \( 1 + (-11.3 - 11.3i)T + 73iT^{2} \)
79 \( 1 + 5.40iT - 79T^{2} \)
83 \( 1 + (8.79 + 8.79i)T + 83iT^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + (-7.76 + 7.76i)T - 97iT^{2} \)
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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

−10.91429839165861065256286603453, −10.02380759325507369182885395655, −8.954141397739901328172769872484, −8.312483403795791373958081174065, −6.96333743303747341006929846426, −6.41361047988139141180470824440, −5.55465885106946879436029634380, −4.50901899404890348036180904589, −3.11712275747352192265827389920, −2.19636878577497436883883748863, 0.981776416218539787518466293209, 2.27253446300269422527995873568, 3.86869290108987300676746215370, 4.55849351196707037529495890379, 5.61022499155591133516860998671, 6.59314167937302232887053537801, 7.61316047283797572949531307412, 8.870190873556390922664783789715, 9.507991150471484196063731664939, 10.42466306195013566236126328642

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