# Properties

 Label 2-35-35.34-c22-0-52 Degree $2$ Conductor $35$ Sign $1$ Analytic cond. $107.347$ Root an. cond. $10.3608$ Motivic weight $22$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 3.41e5·3-s + 4.19e6·4-s − 4.88e7·5-s + 1.97e9·7-s + 8.51e10·9-s + 3.54e11·11-s − 1.43e12·12-s + 1.43e12·13-s + 1.66e13·15-s + 1.75e13·16-s + 6.65e13·17-s − 2.04e14·20-s − 6.74e14·21-s + 2.38e15·25-s − 1.83e16·27-s + 8.29e15·28-s + 2.37e16·29-s − 1.21e17·33-s − 9.65e16·35-s + 3.57e17·36-s − 4.88e17·39-s + 1.48e18·44-s − 4.15e18·45-s − 2.60e18·47-s − 6.00e18·48-s + 3.90e18·49-s − 2.27e19·51-s + ⋯
 L(s)  = 1 − 1.92·3-s + 4-s − 5-s + 7-s + 2.71·9-s + 1.24·11-s − 1.92·12-s + 0.798·13-s + 1.92·15-s + 16-s + 1.94·17-s − 20-s − 1.92·21-s + 25-s − 3.30·27-s + 28-s + 1.94·29-s − 2.39·33-s − 35-s + 2.71·36-s − 1.53·39-s + 1.24·44-s − 2.71·45-s − 1.05·47-s − 1.92·48-s + 49-s − 3.74·51-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(23-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$35$$    =    $$5 \cdot 7$$ Sign: $1$ Analytic conductor: $$107.347$$ Root analytic conductor: $$10.3608$$ Motivic weight: $$22$$ Rational: yes Arithmetic: yes Character: $\chi_{35} (34, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 35,\ (\ :11),\ 1)$$

## Particular Values

 $$L(\frac{23}{2})$$ $$\approx$$ $$2.061648937$$ $$L(\frac12)$$ $$\approx$$ $$2.061648937$$ $$L(12)$$ 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^{11} T$$
7 $$1 - p^{11} T$$
good2 $$( 1 - p^{11} T )( 1 + p^{11} T )$$
3 $$1 + 341351 T + p^{22} T^{2}$$
11 $$1 - 354730232987 T + p^{22} T^{2}$$
13 $$1 - 1431532005269 T + p^{22} T^{2}$$
17 $$1 - 66547133948621 T + p^{22} T^{2}$$
19 $$( 1 - p^{11} T )( 1 + p^{11} T )$$
23 $$( 1 - p^{11} T )( 1 + p^{11} T )$$
29 $$1 - 23774726872835423 T + p^{22} T^{2}$$
31 $$( 1 - p^{11} T )( 1 + p^{11} T )$$
37 $$( 1 - p^{11} T )( 1 + p^{11} T )$$
41 $$( 1 - p^{11} T )( 1 + p^{11} T )$$
43 $$( 1 - p^{11} T )( 1 + p^{11} T )$$
47 $$1 + 2606499276897091519 T + p^{22} T^{2}$$
53 $$( 1 - p^{11} T )( 1 + p^{11} T )$$
59 $$( 1 - p^{11} T )( 1 + p^{11} T )$$
61 $$( 1 - p^{11} T )( 1 + p^{11} T )$$
67 $$( 1 - p^{11} T )( 1 + p^{11} T )$$
71 $$1 + 71331542064478634398 T + p^{22} T^{2}$$
73 $$1 -$$$$33\!\cdots\!34$$$$T + p^{22} T^{2}$$
79 $$1 +$$$$48\!\cdots\!57$$$$T + p^{22} T^{2}$$
83 $$1 -$$$$74\!\cdots\!14$$$$T + p^{22} T^{2}$$
89 $$( 1 - p^{11} T )( 1 + p^{11} T )$$
97 $$1 -$$$$29\!\cdots\!01$$$$T + p^{22} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$