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Properties

Label 2-260-65.8-c1-0-2
Degree $2$
Conductor $260$
Sign $0.550 - 0.835i$
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  + (2.13 + 2.13i)3-s + (−0.666 − 2.13i)5-s + 2.84i·7-s + 6.11i·9-s + (2.66 − 2.66i)11-s + (−0.134 + 3.60i)13-s + (3.13 − 5.97i)15-s + (−2.80 − 2.80i)17-s + (3.97 − 3.97i)19-s + (−6.06 + 6.06i)21-s + (−3.66 + 3.66i)23-s + (−4.11 + 2.84i)25-s + (−6.64 + 6.64i)27-s − 8.17i·29-s + (−3.60 − 3.60i)31-s + ⋯
L(s)  = 1  + (1.23 + 1.23i)3-s + (−0.297 − 0.954i)5-s + 1.07i·7-s + 2.03i·9-s + (0.803 − 0.803i)11-s + (−0.0373 + 0.999i)13-s + (0.809 − 1.54i)15-s + (−0.679 − 0.679i)17-s + (0.912 − 0.912i)19-s + (−1.32 + 1.32i)21-s + (−0.764 + 0.764i)23-s + (−0.822 + 0.568i)25-s + (−1.27 + 1.27i)27-s − 1.51i·29-s + (−0.647 − 0.647i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.550 - 0.835i$
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: \((2,\ 260,\ (\ :1/2),\ 0.550 - 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53922 + 0.829290i\)
\(L(\frac12)\) \(\approx\) \(1.53922 + 0.829290i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.666 + 2.13i)T \)
13 \( 1 + (0.134 - 3.60i)T \)
good3 \( 1 + (-2.13 - 2.13i)T + 3iT^{2} \)
7 \( 1 - 2.84iT - 7T^{2} \)
11 \( 1 + (-2.66 + 2.66i)T - 11iT^{2} \)
17 \( 1 + (2.80 + 2.80i)T + 17iT^{2} \)
19 \( 1 + (-3.97 + 3.97i)T - 19iT^{2} \)
23 \( 1 + (3.66 - 3.66i)T - 23iT^{2} \)
29 \( 1 + 8.17iT - 29T^{2} \)
31 \( 1 + (3.60 + 3.60i)T + 31iT^{2} \)
37 \( 1 + 3.11iT - 37T^{2} \)
41 \( 1 + (-3.64 - 3.64i)T + 41iT^{2} \)
43 \( 1 + (-0.998 + 0.998i)T - 43iT^{2} \)
47 \( 1 + 9.46iT - 47T^{2} \)
53 \( 1 + (-2.78 - 2.78i)T + 53iT^{2} \)
59 \( 1 + (-3.97 - 3.97i)T + 59iT^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 - 0.574T + 67T^{2} \)
71 \( 1 + (4.73 + 4.73i)T + 71iT^{2} \)
73 \( 1 + 8.92T + 73T^{2} \)
79 \( 1 - 5.51iT - 79T^{2} \)
83 \( 1 - 2.75iT - 83T^{2} \)
89 \( 1 + (3.36 + 3.36i)T + 89iT^{2} \)
97 \( 1 - 9.12T + 97T^{2} \)
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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

−11.82726901044774525949984276688, −11.40522616751648903842152614023, −9.715956155064285931961602582253, −9.071314030852891574238368691037, −8.809109123864574543532898001442, −7.61502528843058019479171621021, −5.77367147711468282176666371347, −4.61191879187748414823018152045, −3.74334118231884518986449846817, −2.33531254163976951545207352373, 1.57310929837113236097200419167, 3.06267358520805638901662311957, 4.00029948847555936878921780999, 6.29225887505223847315690596452, 7.20411677225607831731160013637, 7.64516621573500372099239920322, 8.680497604246658351104282644994, 9.951711779353733558736807692149, 10.79788895193381984073543866728, 12.17612819632759324203707175572

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