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Properties

Label 2-260-65.4-c1-0-3
Degree $2$
Conductor $260$
Sign $0.856 - 0.516i$
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  + (2.86 + 1.65i)3-s + (−0.877 − 2.05i)5-s + (0.517 + 0.895i)7-s + (3.96 + 6.86i)9-s + (−2.96 − 1.70i)11-s + (3.57 − 0.455i)13-s + (0.888 − 7.33i)15-s + (2.07 − 1.19i)17-s + (−5.37 + 3.10i)19-s + 3.41i·21-s + (−6.28 − 3.62i)23-s + (−3.46 + 3.60i)25-s + 16.3i·27-s + (0.902 − 1.56i)29-s − 5.80i·31-s + ⋯
L(s)  = 1  + (1.65 + 0.954i)3-s + (−0.392 − 0.919i)5-s + (0.195 + 0.338i)7-s + (1.32 + 2.28i)9-s + (−0.892 − 0.515i)11-s + (0.992 − 0.126i)13-s + (0.229 − 1.89i)15-s + (0.503 − 0.290i)17-s + (−1.23 + 0.711i)19-s + 0.746i·21-s + (−1.31 − 0.756i)23-s + (−0.692 + 0.721i)25-s + 3.13i·27-s + (0.167 − 0.290i)29-s − 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.856 - 0.516i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.856 - 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86373 + 0.518319i\)
\(L(\frac12)\) \(\approx\) \(1.86373 + 0.518319i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.877 + 2.05i)T \)
13 \( 1 + (-3.57 + 0.455i)T \)
good3 \( 1 + (-2.86 - 1.65i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.517 - 0.895i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.96 + 1.70i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.07 + 1.19i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.37 - 3.10i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.28 + 3.62i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.902 + 1.56i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.80iT - 31T^{2} \)
37 \( 1 + (0.713 - 1.23i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.60 + 2.07i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.86 + 1.07i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.50T + 47T^{2} \)
53 \( 1 - 4.55iT - 53T^{2} \)
59 \( 1 + (5.06 - 2.92i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.90 + 3.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.80 - 6.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.49 + 5.48i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.15T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + 0.706T + 83T^{2} \)
89 \( 1 + (-5.06 - 2.92i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.99 - 13.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.23206706897292747472762038769, −10.80837200856655864320142488417, −10.02940138793581300928956632388, −8.953521615203875195479853216980, −8.280922151559762340625315882738, −7.88390536303648778778662712163, −5.71656832446885413832861331612, −4.44879033973867270278607112411, −3.62785138830736830065728296807, −2.21303557239785542591104159617, 1.89251420809459894046738246732, 3.07774402082743044644036825962, 4.07497959922960651495558330575, 6.31711625305607031790409357109, 7.23849410383692504257489107305, 7.952369410481928264493714865402, 8.678566989713181721679920363082, 9.943937455898745705753911026602, 10.85594063465423350708041407339, 12.17865437017321543925267380794

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