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Properties

Label 2-630-35.27-c1-0-3
Degree $2$
Conductor $630$
Sign $0.836 - 0.548i$
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 + (−2.20 + 0.386i)5-s + (−1.57 − 2.12i)7-s + (0.707 − 0.707i)8-s + (1.83 + 1.28i)10-s − 1.41·11-s + (3.66 + 3.66i)13-s + (−0.386 + 2.61i)14-s − 1.00·16-s + (−2.58 + 2.58i)17-s + 4.75·19-s + (−0.386 − 2.20i)20-s + (1.00 + 1.00i)22-s + (3.82 − 3.82i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.984 + 0.172i)5-s + (−0.596 − 0.802i)7-s + (0.250 − 0.250i)8-s + (0.578 + 0.406i)10-s − 0.426·11-s + (1.01 + 1.01i)13-s + (−0.103 + 0.699i)14-s − 0.250·16-s + (−0.627 + 0.627i)17-s + 1.09·19-s + (−0.0863 − 0.492i)20-s + (0.213 + 0.213i)22-s + (0.796 − 0.796i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.836 - 0.548i$
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: \((2,\ 630,\ (\ :1/2),\ 0.836 - 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.710427 + 0.212260i\)
\(L(\frac12)\) \(\approx\) \(0.710427 + 0.212260i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.20 - 0.386i)T \)
7 \( 1 + (1.57 + 2.12i)T \)
good11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 + (-3.66 - 3.66i)T + 13iT^{2} \)
17 \( 1 + (2.58 - 2.58i)T - 17iT^{2} \)
19 \( 1 - 4.75T + 19T^{2} \)
23 \( 1 + (-3.82 + 3.82i)T - 23iT^{2} \)
29 \( 1 - 8.06iT - 29T^{2} \)
31 \( 1 - 9.89iT - 31T^{2} \)
37 \( 1 + (0.701 + 0.701i)T + 37iT^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 + (2 - 2i)T - 43iT^{2} \)
47 \( 1 + (-1.54 + 1.54i)T - 47iT^{2} \)
53 \( 1 + (-6.64 + 6.64i)T - 53iT^{2} \)
59 \( 1 - 7.49T + 59T^{2} \)
61 \( 1 + 6.22iT - 61T^{2} \)
67 \( 1 + (4 + 4i)T + 67iT^{2} \)
71 \( 1 + 0.992T + 71T^{2} \)
73 \( 1 + (-4.59 - 4.59i)T + 73iT^{2} \)
79 \( 1 - 7.40iT - 79T^{2} \)
83 \( 1 + (-8.03 - 8.03i)T + 83iT^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + (1.63 - 1.63i)T - 97iT^{2} \)
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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

−10.89090394508517623715568574296, −9.928820378047852004180257247291, −8.848114748404514515803284835726, −8.266801330256644690473391276679, −7.04622821221035935592751298579, −6.68203108120530179322359344559, −4.86581153364774762474621343931, −3.80869008081459913755016801318, −3.07427328326608007577432933868, −1.19320663677608493946074378678, 0.57047264862290522952290157137, 2.70181697927075912763681689882, 3.86244366834436947576351353805, 5.27896548062392244626358033157, 5.96017051709942134344339722532, 7.20919821970434247487256527671, 7.85894274684009421537152510234, 8.749938472774883931370833808471, 9.409592511494025685023200510434, 10.43804011418008708254274867856

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