Properties

 Label 2-4704-1.1-c1-0-60 Degree $2$ Conductor $4704$ Sign $-1$ Analytic cond. $37.5616$ Root an. cond. $6.12875$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

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Dirichlet series

 L(s)  = 1 + 3-s − 3.04·5-s + 9-s + 3.93·11-s − 4.88·13-s − 3.04·15-s + 5.34·17-s − 2.30·19-s − 7.93·23-s + 4.25·25-s + 27-s + 5.55·29-s − 0.645·31-s + 3.93·33-s + 5.65·37-s − 4.88·39-s − 10.0·41-s + 8.91·43-s − 3.04·45-s − 6.61·47-s + 5.34·51-s + 1.25·53-s − 11.9·55-s − 2.30·57-s + 3.04·59-s + 2.97·61-s + 14.8·65-s + ⋯
 L(s)  = 1 + 0.577·3-s − 1.36·5-s + 0.333·9-s + 1.18·11-s − 1.35·13-s − 0.785·15-s + 1.29·17-s − 0.528·19-s − 1.65·23-s + 0.851·25-s + 0.192·27-s + 1.03·29-s − 0.115·31-s + 0.684·33-s + 0.929·37-s − 0.782·39-s − 1.57·41-s + 1.35·43-s − 0.453·45-s − 0.964·47-s + 0.748·51-s + 0.172·53-s − 1.61·55-s − 0.304·57-s + 0.396·59-s + 0.380·61-s + 1.84·65-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$4704$$    =    $$2^{5} \cdot 3 \cdot 7^{2}$$ Sign: $-1$ Analytic conductor: $$37.5616$$ Root analytic conductor: $$6.12875$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{4704} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 4704,\ (\ :1/2),\ -1)$$

Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - T$$
7 $$1$$
good5 $$1 + 3.04T + 5T^{2}$$
11 $$1 - 3.93T + 11T^{2}$$
13 $$1 + 4.88T + 13T^{2}$$
17 $$1 - 5.34T + 17T^{2}$$
19 $$1 + 2.30T + 19T^{2}$$
23 $$1 + 7.93T + 23T^{2}$$
29 $$1 - 5.55T + 29T^{2}$$
31 $$1 + 0.645T + 31T^{2}$$
37 $$1 - 5.65T + 37T^{2}$$
41 $$1 + 10.0T + 41T^{2}$$
43 $$1 - 8.91T + 43T^{2}$$
47 $$1 + 6.61T + 47T^{2}$$
53 $$1 - 1.25T + 53T^{2}$$
59 $$1 - 3.04T + 59T^{2}$$
61 $$1 - 2.97T + 61T^{2}$$
67 $$1 + 13.5T + 67T^{2}$$
71 $$1 + 13.5T + 71T^{2}$$
73 $$1 + 4.67T + 73T^{2}$$
79 $$1 + 1.05T + 79T^{2}$$
83 $$1 - 8.60T + 83T^{2}$$
89 $$1 + 4.85T + 89T^{2}$$
97 $$1 + 18.3T + 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

−7.958347437055312632402063967166, −7.39486043678060150761823143236, −6.70178983050122089305005954147, −5.79228417998789422882098580441, −4.63952369988187650605023347117, −4.13169057056592715118058325792, −3.44367213109694849847679642344, −2.56443177363923014464589155469, −1.36612095705292655408578235408, 0, 1.36612095705292655408578235408, 2.56443177363923014464589155469, 3.44367213109694849847679642344, 4.13169057056592715118058325792, 4.63952369988187650605023347117, 5.79228417998789422882098580441, 6.70178983050122089305005954147, 7.39486043678060150761823143236, 7.958347437055312632402063967166