# Properties

 Label 2-260-260.207-c1-0-21 Degree $2$ Conductor $260$ Sign $-0.339 + 0.940i$ Analytic cond. $2.07611$ Root an. cond. $1.44087$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.23 + 0.693i)2-s + (−0.828 − 0.828i)3-s + (1.03 − 1.70i)4-s + (−1.31 − 1.80i)5-s + (1.59 + 0.446i)6-s + (2.94 + 2.94i)7-s + (−0.0923 + 2.82i)8-s − 1.62i·9-s + (2.87 + 1.30i)10-s − 3.40·11-s + (−2.27 + 0.556i)12-s + (−1.99 − 3.00i)13-s + (−5.66 − 1.58i)14-s + (−0.402 + 2.58i)15-s + (−1.84 − 3.54i)16-s + (−2.96 − 2.96i)17-s + ⋯
 L(s)  = 1 + (−0.871 + 0.490i)2-s + (−0.478 − 0.478i)3-s + (0.518 − 0.854i)4-s + (−0.590 − 0.807i)5-s + (0.651 + 0.182i)6-s + (1.11 + 1.11i)7-s + (−0.0326 + 0.999i)8-s − 0.542i·9-s + (0.910 + 0.414i)10-s − 1.02·11-s + (−0.656 + 0.160i)12-s + (−0.552 − 0.833i)13-s + (−1.51 − 0.423i)14-s + (−0.103 + 0.668i)15-s + (−0.461 − 0.886i)16-s + (−0.719 − 0.719i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.339 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$260$$    =    $$2^{2} \cdot 5 \cdot 13$$ Sign: $-0.339 + 0.940i$ Analytic conductor: $$2.07611$$ Root analytic conductor: $$1.44087$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{260} (207, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 260,\ (\ :1/2),\ -0.339 + 0.940i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.268811 - 0.382812i$$ $$L(\frac12)$$ $$\approx$$ $$0.268811 - 0.382812i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.23 - 0.693i)T$$
5 $$1 + (1.31 + 1.80i)T$$
13 $$1 + (1.99 + 3.00i)T$$
good3 $$1 + (0.828 + 0.828i)T + 3iT^{2}$$
7 $$1 + (-2.94 - 2.94i)T + 7iT^{2}$$
11 $$1 + 3.40T + 11T^{2}$$
17 $$1 + (2.96 + 2.96i)T + 17iT^{2}$$
19 $$1 + 7.33iT - 19T^{2}$$
23 $$1 + (-1.65 - 1.65i)T + 23iT^{2}$$
29 $$1 + 1.98iT - 29T^{2}$$
31 $$1 + 4.43T + 31T^{2}$$
37 $$1 + (-1.88 + 1.88i)T - 37iT^{2}$$
41 $$1 + 7.55iT - 41T^{2}$$
43 $$1 + (-0.548 - 0.548i)T + 43iT^{2}$$
47 $$1 + (5.58 + 5.58i)T + 47iT^{2}$$
53 $$1 + (0.437 - 0.437i)T - 53iT^{2}$$
59 $$1 - 6.14iT - 59T^{2}$$
61 $$1 - 8.12T + 61T^{2}$$
67 $$1 + (-3.78 - 3.78i)T + 67iT^{2}$$
71 $$1 - 8.90T + 71T^{2}$$
73 $$1 + (-4.20 - 4.20i)T + 73iT^{2}$$
79 $$1 - 13.2T + 79T^{2}$$
83 $$1 + (0.688 - 0.688i)T - 83iT^{2}$$
89 $$1 - 17.5T + 89T^{2}$$
97 $$1 + (3.63 - 3.63i)T - 97iT^{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

−11.52872942286243473080746192762, −11.00258263925892552631035401969, −9.410523024085175566674575519534, −8.724858617474228105656879263381, −7.83409709081385328231737062722, −6.99898385496930966818841337965, −5.42128656443919921205684316237, −5.06627462909974047919701846296, −2.35695594652149723470277505641, −0.49506960848957732323250687056, 2.02725156100717459431274891057, 3.79273450411814582949833001450, 4.77377289733053947479211254920, 6.61965146935936234273168959976, 7.81404328952439901260640927964, 8.047898717687090722409867359986, 9.813062583147819881240355904656, 10.73487251178194166511584057613, 10.87153786188879683427269367056, 11.77338943215927932513634628223