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Properties

Label 2-260-20.7-c1-0-17
Degree $2$
Conductor $260$
Sign $-0.709 + 0.704i$
Analytic cond. $2.07611$
Root an. cond. $1.44087$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 − 0.0788i)2-s + (−1.63 − 1.63i)3-s + (1.98 + 0.222i)4-s + (−0.963 + 2.01i)5-s + (2.17 + 2.43i)6-s + (3.04 − 3.04i)7-s + (−2.78 − 0.470i)8-s + 2.33i·9-s + (1.51 − 2.77i)10-s − 0.497i·11-s + (−2.88 − 3.60i)12-s + (0.707 − 0.707i)13-s + (−4.54 + 4.06i)14-s + (4.86 − 1.72i)15-s + (3.90 + 0.884i)16-s + (−2.41 − 2.41i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0557i)2-s + (−0.942 − 0.942i)3-s + (0.993 + 0.111i)4-s + (−0.430 + 0.902i)5-s + (0.888 + 0.993i)6-s + (1.15 − 1.15i)7-s + (−0.986 − 0.166i)8-s + 0.777i·9-s + (0.480 − 0.876i)10-s − 0.149i·11-s + (−0.832 − 1.04i)12-s + (0.196 − 0.196i)13-s + (−1.21 + 1.08i)14-s + (1.25 − 0.444i)15-s + (0.975 + 0.221i)16-s + (−0.585 − 0.585i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.709 + 0.704i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.709 + 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.168116 - 0.407885i\)
\(L(\frac12)\) \(\approx\) \(0.168116 - 0.407885i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0788i)T \)
5 \( 1 + (0.963 - 2.01i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (1.63 + 1.63i)T + 3iT^{2} \)
7 \( 1 + (-3.04 + 3.04i)T - 7iT^{2} \)
11 \( 1 + 0.497iT - 11T^{2} \)
17 \( 1 + (2.41 + 2.41i)T + 17iT^{2} \)
19 \( 1 + 5.62T + 19T^{2} \)
23 \( 1 + (4.41 + 4.41i)T + 23iT^{2} \)
29 \( 1 + 5.13iT - 29T^{2} \)
31 \( 1 + 3.14iT - 31T^{2} \)
37 \( 1 + (-4.40 - 4.40i)T + 37iT^{2} \)
41 \( 1 - 8.49T + 41T^{2} \)
43 \( 1 + (5.16 + 5.16i)T + 43iT^{2} \)
47 \( 1 + (-0.110 + 0.110i)T - 47iT^{2} \)
53 \( 1 + (-0.203 + 0.203i)T - 53iT^{2} \)
59 \( 1 - 7.36T + 59T^{2} \)
61 \( 1 + 6.43T + 61T^{2} \)
67 \( 1 + (-7.40 + 7.40i)T - 67iT^{2} \)
71 \( 1 - 0.618iT - 71T^{2} \)
73 \( 1 + (11.5 - 11.5i)T - 73iT^{2} \)
79 \( 1 - 2.69T + 79T^{2} \)
83 \( 1 + (-5.93 - 5.93i)T + 83iT^{2} \)
89 \( 1 + 12.7iT - 89T^{2} \)
97 \( 1 + (-8.15 - 8.15i)T + 97iT^{2} \)
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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

−11.30143235941395072073512923025, −10.94795586807295589276349035011, −10.07592124825778736723925279156, −8.331230540886836037867842591870, −7.65061081206718815427887382494, −6.83003069224029905580534261131, −6.10025572636240126071667894260, −4.20061917964375469934322567879, −2.17595870875327261197273031389, −0.52241506492675401546837511837, 1.85834868544381851834580521351, 4.24260997598854481303996848466, 5.28975693638546555931890132761, 6.13346073877390701601760546948, 7.84438023544341247888881133007, 8.647105580782071901579463709805, 9.344938692008502493000468633905, 10.57247858345058964601649685859, 11.28819522893380197805050459455, 11.87297954224133819244275529652

Graph of the $Z$-function along the critical line