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Properties

Label 2-260-260.103-c1-0-10
Degree $2$
Conductor $260$
Sign $-0.172 - 0.985i$
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  + (−0.528 + 1.31i)2-s + (0.834 − 0.834i)3-s + (−1.44 − 1.38i)4-s + (0.462 + 2.18i)5-s + (0.653 + 1.53i)6-s + (−1.21 + 1.21i)7-s + (2.58 − 1.15i)8-s + 1.60i·9-s + (−3.11 − 0.549i)10-s + 0.135·11-s + (−2.35 + 0.0453i)12-s + (3.45 + 1.01i)13-s + (−0.948 − 2.22i)14-s + (2.21 + 1.43i)15-s + (0.153 + 3.99i)16-s + (−1.10 + 1.10i)17-s + ⋯
L(s)  = 1  + (−0.373 + 0.927i)2-s + (0.481 − 0.481i)3-s + (−0.720 − 0.693i)4-s + (0.206 + 0.978i)5-s + (0.266 + 0.626i)6-s + (−0.457 + 0.457i)7-s + (0.912 − 0.409i)8-s + 0.535i·9-s + (−0.984 − 0.173i)10-s + 0.0409·11-s + (−0.681 + 0.0130i)12-s + (0.959 + 0.282i)13-s + (−0.253 − 0.595i)14-s + (0.570 + 0.371i)15-s + (0.0384 + 0.999i)16-s + (−0.269 + 0.269i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.172 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.172 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.172 - 0.985i$
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: \((2,\ 260,\ (\ :1/2),\ -0.172 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.720761 + 0.857816i\)
\(L(\frac12)\) \(\approx\) \(0.720761 + 0.857816i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.528 - 1.31i)T \)
5 \( 1 + (-0.462 - 2.18i)T \)
13 \( 1 + (-3.45 - 1.01i)T \)
good3 \( 1 + (-0.834 + 0.834i)T - 3iT^{2} \)
7 \( 1 + (1.21 - 1.21i)T - 7iT^{2} \)
11 \( 1 - 0.135T + 11T^{2} \)
17 \( 1 + (1.10 - 1.10i)T - 17iT^{2} \)
19 \( 1 - 6.12iT - 19T^{2} \)
23 \( 1 + (-4.72 + 4.72i)T - 23iT^{2} \)
29 \( 1 + 4.39iT - 29T^{2} \)
31 \( 1 + 3.12T + 31T^{2} \)
37 \( 1 + (6.39 + 6.39i)T + 37iT^{2} \)
41 \( 1 + 2.92iT - 41T^{2} \)
43 \( 1 + (-1.51 + 1.51i)T - 43iT^{2} \)
47 \( 1 + (-8.14 + 8.14i)T - 47iT^{2} \)
53 \( 1 + (5.39 + 5.39i)T + 53iT^{2} \)
59 \( 1 - 5.47iT - 59T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 + (-7.10 + 7.10i)T - 67iT^{2} \)
71 \( 1 + 4.49T + 71T^{2} \)
73 \( 1 + (-6.04 + 6.04i)T - 73iT^{2} \)
79 \( 1 - 0.944T + 79T^{2} \)
83 \( 1 + (4.61 + 4.61i)T + 83iT^{2} \)
89 \( 1 - 3.22T + 89T^{2} \)
97 \( 1 + (-2.31 - 2.31i)T + 97iT^{2} \)
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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

−12.53438963563908982366374273932, −10.95258412574558236019145937735, −10.29048502018555982443480419567, −9.087388344577539064493483905969, −8.293867663182661628174830510450, −7.28656433259499040945584002115, −6.45019101742036847361520597805, −5.55112460847621318663788100279, −3.75339062043901469758444568344, −2.06735596505652320299128355632, 1.05048425158627131797961211843, 3.04581125133299582235114482948, 4.03726369158667298787216221560, 5.17881589215989459926539955516, 6.92000078784677528450280121765, 8.362519957312162755595407045080, 9.115033635141083731780421945925, 9.589456717121723972355762356784, 10.73036502337786275922550238560, 11.60491700012237413234481902524

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