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Properties

Label 2-260-65.8-c1-0-0
Degree $2$
Conductor $260$
Sign $0.939 - 0.342i$
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  + (−1.42 − 1.42i)3-s + (1.72 + 1.42i)5-s + 4.91i·7-s + 1.06i·9-s + (0.276 − 0.276i)11-s + (3.42 − 1.12i)13-s + (−0.424 − 4.48i)15-s + (3.14 + 3.14i)17-s + (2.48 − 2.48i)19-s + (6.99 − 6.99i)21-s + (−1.27 + 1.27i)23-s + (0.938 + 4.91i)25-s + (−2.76 + 2.76i)27-s − 5.46i·29-s + (1.12 + 1.12i)31-s + ⋯
L(s)  = 1  + (−0.822 − 0.822i)3-s + (0.770 + 0.637i)5-s + 1.85i·7-s + 0.353i·9-s + (0.0834 − 0.0834i)11-s + (0.949 − 0.312i)13-s + (−0.109 − 1.15i)15-s + (0.763 + 0.763i)17-s + (0.570 − 0.570i)19-s + (1.52 − 1.52i)21-s + (−0.266 + 0.266i)23-s + (0.187 + 0.982i)25-s + (−0.531 + 0.531i)27-s − 1.01i·29-s + (0.202 + 0.202i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10676 + 0.195741i\)
\(L(\frac12)\) \(\approx\) \(1.10676 + 0.195741i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 1.42i)T \)
13 \( 1 + (-3.42 + 1.12i)T \)
good3 \( 1 + (1.42 + 1.42i)T + 3iT^{2} \)
7 \( 1 - 4.91iT - 7T^{2} \)
11 \( 1 + (-0.276 + 0.276i)T - 11iT^{2} \)
17 \( 1 + (-3.14 - 3.14i)T + 17iT^{2} \)
19 \( 1 + (-2.48 + 2.48i)T - 19iT^{2} \)
23 \( 1 + (1.27 - 1.27i)T - 23iT^{2} \)
29 \( 1 + 5.46iT - 29T^{2} \)
31 \( 1 + (-1.12 - 1.12i)T + 31iT^{2} \)
37 \( 1 - 1.93iT - 37T^{2} \)
41 \( 1 + (0.237 + 0.237i)T + 41iT^{2} \)
43 \( 1 + (6.16 - 6.16i)T - 43iT^{2} \)
47 \( 1 + 13.3iT - 47T^{2} \)
53 \( 1 + (-2.50 - 2.50i)T + 53iT^{2} \)
59 \( 1 + (-2.48 - 2.48i)T + 59iT^{2} \)
61 \( 1 + 1.25T + 61T^{2} \)
67 \( 1 - 9.76T + 67T^{2} \)
71 \( 1 + (3.61 + 3.61i)T + 71iT^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + 8.07iT - 79T^{2} \)
83 \( 1 + 11.2iT - 83T^{2} \)
89 \( 1 + (-8.16 - 8.16i)T + 89iT^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−11.88888371686150879481179947345, −11.45619459524647661034287513528, −10.20306736409358524936832903021, −9.168405995114640729375155949193, −8.142584131919569121500047844142, −6.72623895422650076075454590624, −5.91600732728809107442965416198, −5.48909518327577227843558348305, −3.12289038825013155740295364735, −1.72616182262416600700108333457, 1.14228121568982728795594409616, 3.73926103348645964961103367730, 4.69644136395765018055023067431, 5.65040418567341855585470419539, 6.82239853866242915105361193499, 8.024446166381580447574271337245, 9.439467501722652380224366828765, 10.16646980948147828087604742173, 10.76920739037190308197033745957, 11.72227724835633028658593825256

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