# Properties

 Label 2-35-35.34-c12-0-25 Degree $2$ Conductor $35$ Sign $1$ Analytic cond. $31.9897$ Root an. cond. $5.65595$ Motivic weight $12$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 782·3-s + 4.09e3·4-s + 1.56e4·5-s + 1.17e5·7-s + 8.00e4·9-s + 2.81e6·11-s − 3.20e6·12-s − 1.95e6·13-s − 1.22e7·15-s + 1.67e7·16-s − 4.77e7·17-s + 6.40e7·20-s − 9.20e7·21-s + 2.44e8·25-s + 3.52e8·27-s + 4.81e8·28-s + 9.13e8·29-s − 2.20e9·33-s + 1.83e9·35-s + 3.28e8·36-s + 1.53e9·39-s + 1.15e10·44-s + 1.25e9·45-s + 9.29e9·47-s − 1.31e10·48-s + 1.38e10·49-s + 3.73e10·51-s + ⋯
 L(s)  = 1 − 1.07·3-s + 4-s + 5-s + 7-s + 0.150·9-s + 1.59·11-s − 1.07·12-s − 0.405·13-s − 1.07·15-s + 16-s − 1.97·17-s + 20-s − 1.07·21-s + 25-s + 0.911·27-s + 28-s + 1.53·29-s − 1.70·33-s + 35-s + 0.150·36-s + 0.435·39-s + 1.59·44-s + 0.150·45-s + 0.862·47-s − 1.07·48-s + 49-s + 2.12·51-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$35$$    =    $$5 \cdot 7$$ Sign: $1$ Analytic conductor: $$31.9897$$ Root analytic conductor: $$5.65595$$ Motivic weight: $$12$$ Rational: yes Arithmetic: yes Character: $\chi_{35} (34, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 35,\ (\ :6),\ 1)$$

## Particular Values

 $$L(\frac{13}{2})$$ $$\approx$$ $$2.500669605$$ $$L(\frac12)$$ $$\approx$$ $$2.500669605$$ $$L(7)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 - p^{6} T$$
7 $$1 - p^{6} T$$
good2 $$( 1 - p^{6} T )( 1 + p^{6} T )$$
3 $$1 + 782 T + p^{12} T^{2}$$
11 $$1 - 2817362 T + p^{12} T^{2}$$
13 $$1 + 1958542 T + p^{12} T^{2}$$
17 $$1 + 47706622 T + p^{12} T^{2}$$
19 $$( 1 - p^{6} T )( 1 + p^{6} T )$$
23 $$( 1 - p^{6} T )( 1 + p^{6} T )$$
29 $$1 - 913676402 T + p^{12} T^{2}$$
31 $$( 1 - p^{6} T )( 1 + p^{6} T )$$
37 $$( 1 - p^{6} T )( 1 + p^{6} T )$$
41 $$( 1 - p^{6} T )( 1 + p^{6} T )$$
43 $$( 1 - p^{6} T )( 1 + p^{6} T )$$
47 $$1 - 9293086658 T + p^{12} T^{2}$$
53 $$( 1 - p^{6} T )( 1 + p^{6} T )$$
59 $$( 1 - p^{6} T )( 1 + p^{6} T )$$
61 $$( 1 - p^{6} T )( 1 + p^{6} T )$$
67 $$( 1 - p^{6} T )( 1 + p^{6} T )$$
71 $$1 + 255286231198 T + p^{12} T^{2}$$
73 $$1 + 48396356062 T + p^{12} T^{2}$$
79 $$1 - 379435754882 T + p^{12} T^{2}$$
83 $$1 - 648698638898 T + p^{12} T^{2}$$
89 $$( 1 - p^{6} T )( 1 + p^{6} T )$$
97 $$1 + 859766289982 T + p^{12} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$