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Properties

Label 2-260-260.103-c1-0-11
Degree $2$
Conductor $260$
Sign $-0.477 - 0.878i$
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 + i)2-s + 2i·4-s + (−2 + i)5-s + (−2 + 2i)8-s + 3i·9-s + (−3 − i)10-s + (2 + 3i)13-s − 4·16-s + (5 − 5i)17-s + (−3 + 3i)18-s + (−2 − 4i)20-s + (3 − 4i)25-s + (−1 + 5i)26-s − 4i·29-s + (−4 − 4i)32-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·4-s + (−0.894 + 0.447i)5-s + (−0.707 + 0.707i)8-s + i·9-s + (−0.948 − 0.316i)10-s + (0.554 + 0.832i)13-s − 16-s + (1.21 − 1.21i)17-s + (−0.707 + 0.707i)18-s + (−0.447 − 0.894i)20-s + (0.600 − 0.800i)25-s + (−0.196 + 0.980i)26-s − 0.742i·29-s + (−0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.477 - 0.878i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.477 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771796 + 1.29824i\)
\(L(\frac12)\) \(\approx\) \(0.771796 + 1.29824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
5 \( 1 + (2 - i)T \)
13 \( 1 + (-2 - 3i)T \)
good3 \( 1 - 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + (-5 + 5i)T - 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-7 - 7i)T + 37iT^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (-5 - 5i)T + 53iT^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-11 + 11i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (-13 - 13i)T + 97iT^{2} \)
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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

−12.19808920026082555490503275323, −11.62520358704486164764320485690, −10.67440659049052817001036729708, −9.197057706338527608000351424781, −7.956562089194112718817643091403, −7.46251951013669094485232831439, −6.33534823329739839119619728991, −5.06211686255983221878751417400, −4.04717583880578270689560921770, −2.78410593719181404868377758581, 1.05784920675631122175991650457, 3.27607928396725695438596259054, 3.99713765636240581374531385895, 5.37054128530512854989016519804, 6.37087300067550339343483220817, 7.83112531043615601525828886727, 8.893088393591458338606294055737, 9.985301004627063300813341812259, 10.94764304595673512092141508531, 11.83479989647058711874851966046

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