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Properties

Label 2-630-21.20-c1-0-14
Degree $2$
Conductor $630$
Sign $-0.606 + 0.795i$
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 − 4-s + 5-s + (0.0951 − 2.64i)7-s + i·8-s − i·10-s − 5.28i·11-s + 2.19i·13-s + (−2.64 − 0.0951i)14-s + 16-s − 1.04·17-s + 6.43i·19-s − 20-s − 5.28·22-s − 7.47i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.447·5-s + (0.0359 − 0.999i)7-s + 0.353i·8-s − 0.316i·10-s − 1.59i·11-s + 0.607i·13-s + (−0.706 − 0.0254i)14-s + 0.250·16-s − 0.253·17-s + 1.47i·19-s − 0.223·20-s − 1.12·22-s − 1.55i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.606 + 0.795i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.606 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.595386 - 1.20272i\)
\(L(\frac12)\) \(\approx\) \(0.595386 - 1.20272i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
7 \( 1 + (-0.0951 + 2.64i)T \)
good11 \( 1 + 5.28iT - 11T^{2} \)
13 \( 1 - 2.19iT - 13T^{2} \)
17 \( 1 + 1.04T + 17T^{2} \)
19 \( 1 - 6.43iT - 19T^{2} \)
23 \( 1 + 7.47iT - 23T^{2} \)
29 \( 1 + 7.47iT - 29T^{2} \)
31 \( 1 + 9.09iT - 31T^{2} \)
37 \( 1 + 0.855T + 37T^{2} \)
41 \( 1 - 2.19T + 41T^{2} \)
43 \( 1 + 0.954T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 - 3.09iT - 53T^{2} \)
59 \( 1 - 13.7T + 59T^{2} \)
61 \( 1 - 8.05iT - 61T^{2} \)
67 \( 1 - 5.33T + 67T^{2} \)
71 \( 1 - 6.43iT - 71T^{2} \)
73 \( 1 - 4.57iT - 73T^{2} \)
79 \( 1 - 15.6T + 79T^{2} \)
83 \( 1 + 4.38T + 83T^{2} \)
89 \( 1 - 4.28T + 89T^{2} \)
97 \( 1 + 11.8iT - 97T^{2} \)
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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

−10.30102577769429263721876589997, −9.701849190428511006931769369835, −8.560249121519800276824684041607, −7.923296666456457294681295401002, −6.52514226843610893096541129052, −5.78130379487304460226700488174, −4.38456394244966442744490622944, −3.61693548872759052948817886281, −2.26095379912379820963609874110, −0.75370136677137658145667636152, 1.85942740762765502977467228977, 3.22392123092965668838638689454, 4.95451551416376751012762600256, 5.22973757618703248394512767830, 6.58284462605950871461682173720, 7.18760356697434900147903027570, 8.297866334986300978208610396227, 9.210780178788708379057280766991, 9.704495569692674095793518788188, 10.79124794755792221647815094373

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