Properties

Label 2-260-13.3-c1-0-0
Degree $2$
Conductor $260$
Sign $0.0128 - 0.999i$
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  + (0.5 + 0.866i)3-s − 5-s + (−2.5 + 4.33i)7-s + (1 − 1.73i)9-s + (2.5 + 4.33i)11-s + (−1 + 3.46i)13-s + (−0.5 − 0.866i)15-s + (0.5 − 0.866i)17-s + (1.5 − 2.59i)19-s − 5·21-s + (−1.5 − 2.59i)23-s + 25-s + 5·27-s + (0.5 + 0.866i)29-s + (−2.5 + 4.33i)33-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s − 0.447·5-s + (−0.944 + 1.63i)7-s + (0.333 − 0.577i)9-s + (0.753 + 1.30i)11-s + (−0.277 + 0.960i)13-s + (−0.129 − 0.223i)15-s + (0.121 − 0.210i)17-s + (0.344 − 0.596i)19-s − 1.09·21-s + (−0.312 − 0.541i)23-s + 0.200·25-s + 0.962·27-s + (0.0928 + 0.160i)29-s + (−0.435 + 0.753i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.822973 + 0.812487i\)
\(L(\frac12)\) \(\approx\) \(0.822973 + 0.812487i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 + (1 - 3.46i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.5 - 4.33i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + (-5.5 + 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.5 - 11.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.5 - 9.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.24924731043351185670842179444, −11.56025846344221747983006624058, −9.931394610781556530995278621074, −9.341937425564374435374481987126, −8.773381552258027316656363259159, −7.12470978442532819385875394780, −6.34250498390718988551963612027, −4.85857126071765306658023207758, −3.74217712127875368808055793406, −2.37637429776689080483134120293, 0.938552208271945488268348878911, 3.21492978616317955927313025929, 4.08004395856611851986330547778, 5.81242186146166551164724814922, 7.04654188718715583010379475782, 7.65241651154255264766333961869, 8.673837985662491270895810324127, 10.13230188100752471625254207499, 10.59128653523405392252581592363, 11.82652505519072324707373203067

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