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Properties

Label 2-260-65.7-c1-0-0
Degree $2$
Conductor $260$
Sign $-0.469 - 0.882i$
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  + (0.674 + 2.51i)3-s + (−0.647 + 2.14i)5-s + (−0.839 − 1.45i)7-s + (−3.29 + 1.90i)9-s + (0.758 + 2.83i)11-s + (3.53 − 0.727i)13-s + (−5.82 − 0.186i)15-s + (−7.90 − 2.11i)17-s + (2.32 + 0.621i)19-s + (3.09 − 3.09i)21-s + (1.57 − 0.421i)23-s + (−4.16 − 2.77i)25-s + (−1.47 − 1.47i)27-s + (2.41 + 1.39i)29-s + (7.32 + 7.32i)31-s + ⋯
L(s)  = 1  + (0.389 + 1.45i)3-s + (−0.289 + 0.957i)5-s + (−0.317 − 0.549i)7-s + (−1.09 + 0.633i)9-s + (0.228 + 0.853i)11-s + (0.979 − 0.201i)13-s + (−1.50 − 0.0480i)15-s + (−1.91 − 0.513i)17-s + (0.532 + 0.142i)19-s + (0.675 − 0.675i)21-s + (0.328 − 0.0879i)23-s + (−0.832 − 0.554i)25-s + (−0.284 − 0.284i)27-s + (0.448 + 0.259i)29-s + (1.31 + 1.31i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.469 - 0.882i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.469 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.657443 + 1.09441i\)
\(L(\frac12)\) \(\approx\) \(0.657443 + 1.09441i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.647 - 2.14i)T \)
13 \( 1 + (-3.53 + 0.727i)T \)
good3 \( 1 + (-0.674 - 2.51i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.839 + 1.45i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.758 - 2.83i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (7.90 + 2.11i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.32 - 0.621i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.57 + 0.421i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.41 - 1.39i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.32 - 7.32i)T + 31iT^{2} \)
37 \( 1 + (-2.08 + 3.60i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.77 + 1.28i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.786 + 2.93i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + (6.66 - 6.66i)T - 53iT^{2} \)
59 \( 1 + (1.16 - 4.35i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.16 + 3.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.28 + 2.47i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.63 + 6.10i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 7.13iT - 73T^{2} \)
79 \( 1 + 9.73iT - 79T^{2} \)
83 \( 1 - 6.67T + 83T^{2} \)
89 \( 1 + (-0.578 + 0.154i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (10.4 - 6.03i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.08450402192835693218764355473, −10.76709991438589976438113495591, −10.65530780119964096765345164977, −9.507947209286489352173350737557, −8.743917735363422959658341072308, −7.33073359778827816611782045952, −6.38815381369955657294775542703, −4.68763441396959567537809453029, −3.89720440611654403554415648343, −2.80689072892184153007011759440, 1.04711940603510689611019257416, 2.56403918919994340483857261456, 4.23099466216416176279850750215, 5.92432463165020916307226317510, 6.60783742917373654299248085626, 7.975822664793409104886996829129, 8.584587749249546127962967039865, 9.301356251646542060853805400732, 11.15611952542192364334435379633, 11.81729473162779023745371388284

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