# Properties

 Label 2-4368-1.1-c1-0-38 Degree $2$ Conductor $4368$ Sign $-1$ Analytic cond. $34.8786$ Root an. cond. $5.90581$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3-s − 3.56·5-s + 7-s + 9-s − 3.12·11-s + 13-s + 3.56·15-s + 1.12·17-s + 1.56·19-s − 21-s + 1.56·23-s + 7.68·25-s − 27-s − 7.56·29-s + 5.56·31-s + 3.12·33-s − 3.56·35-s + 1.12·37-s − 39-s − 2·41-s + 1.56·43-s − 3.56·45-s + 8.68·47-s + 49-s − 1.12·51-s − 1.31·53-s + 11.1·55-s + ⋯
 L(s)  = 1 − 0.577·3-s − 1.59·5-s + 0.377·7-s + 0.333·9-s − 0.941·11-s + 0.277·13-s + 0.919·15-s + 0.272·17-s + 0.358·19-s − 0.218·21-s + 0.325·23-s + 1.53·25-s − 0.192·27-s − 1.40·29-s + 0.998·31-s + 0.543·33-s − 0.602·35-s + 0.184·37-s − 0.160·39-s − 0.312·41-s + 0.238·43-s − 0.530·45-s + 1.26·47-s + 0.142·49-s − 0.157·51-s − 0.180·53-s + 1.49·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4368$$    =    $$2^{4} \cdot 3 \cdot 7 \cdot 13$$ Sign: $-1$ Analytic conductor: $$34.8786$$ Root analytic conductor: $$5.90581$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 4368,\ (\ :1/2),\ -1)$$

## 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 - T$$
13 $$1 - T$$
good5 $$1 + 3.56T + 5T^{2}$$
11 $$1 + 3.12T + 11T^{2}$$
17 $$1 - 1.12T + 17T^{2}$$
19 $$1 - 1.56T + 19T^{2}$$
23 $$1 - 1.56T + 23T^{2}$$
29 $$1 + 7.56T + 29T^{2}$$
31 $$1 - 5.56T + 31T^{2}$$
37 $$1 - 1.12T + 37T^{2}$$
41 $$1 + 2T + 41T^{2}$$
43 $$1 - 1.56T + 43T^{2}$$
47 $$1 - 8.68T + 47T^{2}$$
53 $$1 + 1.31T + 53T^{2}$$
59 $$1 + 2.24T + 59T^{2}$$
61 $$1 - 6T + 61T^{2}$$
67 $$1 + 0.876T + 67T^{2}$$
71 $$1 - 10.2T + 71T^{2}$$
73 $$1 + 0.438T + 73T^{2}$$
79 $$1 + 0.684T + 79T^{2}$$
83 $$1 - 12.6T + 83T^{2}$$
89 $$1 + 10.6T + 89T^{2}$$
97 $$1 + 14.6T + 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

−7.87575271323689612241316906679, −7.44667140030201639559076286664, −6.68739668382502249755221281317, −5.62690127594722911619964936076, −5.03209765246110182136305336756, −4.20616277974553578246526970616, −3.56315743509505356394725820019, −2.54677968075550301463800410797, −1.08444517067964097977084642172, 0, 1.08444517067964097977084642172, 2.54677968075550301463800410797, 3.56315743509505356394725820019, 4.20616277974553578246526970616, 5.03209765246110182136305336756, 5.62690127594722911619964936076, 6.68739668382502249755221281317, 7.44667140030201639559076286664, 7.87575271323689612241316906679