# Properties

 Label 2-311696-1.1-c1-0-25 Degree $2$ Conductor $311696$ Sign $-1$ Analytic cond. $2488.90$ Root an. cond. $49.8889$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 2·3-s + 7-s + 9-s − 6·17-s − 6·19-s − 2·21-s + 23-s − 5·25-s + 4·27-s − 10·29-s − 4·31-s − 2·37-s + 10·41-s − 4·43-s − 12·47-s + 49-s + 12·51-s − 6·53-s + 12·57-s + 2·59-s + 63-s − 2·69-s + 8·71-s + 6·73-s + 10·75-s − 8·79-s − 11·81-s + ⋯
 L(s)  = 1 − 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.45·17-s − 1.37·19-s − 0.436·21-s + 0.208·23-s − 25-s + 0.769·27-s − 1.85·29-s − 0.718·31-s − 0.328·37-s + 1.56·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 1.68·51-s − 0.824·53-s + 1.58·57-s + 0.260·59-s + 0.125·63-s − 0.240·69-s + 0.949·71-s + 0.702·73-s + 1.15·75-s − 0.900·79-s − 1.22·81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 311696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 311696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$311696$$    =    $$2^{4} \cdot 7 \cdot 11^{2} \cdot 23$$ Sign: $-1$ Analytic conductor: $$2488.90$$ Root analytic conductor: $$49.8889$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{311696} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 311696,\ (\ :1/2),\ -1)$$

## 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$$
7 $$1 - T$$
11 $$1$$
23 $$1 - T$$
good3 $$1 + 2 T + p T^{2}$$
5 $$1 + p T^{2}$$
13 $$1 + p T^{2}$$
17 $$1 + 6 T + p T^{2}$$
19 $$1 + 6 T + p T^{2}$$
29 $$1 + 10 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 + 2 T + p T^{2}$$
41 $$1 - 10 T + p T^{2}$$
43 $$1 + 4 T + p T^{2}$$
47 $$1 + 12 T + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 - 2 T + p T^{2}$$
61 $$1 + p T^{2}$$
67 $$1 + p T^{2}$$
71 $$1 - 8 T + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 + 8 T + p T^{2}$$
83 $$1 + 14 T + p T^{2}$$
89 $$1 + 14 T + p T^{2}$$
97 $$1 + 2 T + p T^{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

−12.79602517209440, −12.57913879600516, −11.77873149756398, −11.27362915570696, −11.19437477269056, −10.89228761066217, −10.27189629559294, −9.662048897837781, −9.332968827533829, −8.584601427885773, −8.430364263635081, −7.738443536355170, −7.122232210687389, −6.819106816051635, −6.183320412033558, −5.863626372532895, −5.447041353179197, −4.714714204447164, −4.510488942043762, −3.888292147667142, −3.329831498737960, −2.443485812704484, −1.957228468987961, −1.518039652185260, −0.4525920594322009, 0, 0.4525920594322009, 1.518039652185260, 1.957228468987961, 2.443485812704484, 3.329831498737960, 3.888292147667142, 4.510488942043762, 4.714714204447164, 5.447041353179197, 5.863626372532895, 6.183320412033558, 6.819106816051635, 7.122232210687389, 7.738443536355170, 8.430364263635081, 8.584601427885773, 9.332968827533829, 9.662048897837781, 10.27189629559294, 10.89228761066217, 11.19437477269056, 11.27362915570696, 11.77873149756398, 12.57913879600516, 12.79602517209440