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Properties

Label 2-261-29.20-c1-0-2
Degree $2$
Conductor $261$
Sign $-0.00819 - 0.999i$
Analytic cond. $2.08409$
Root an. cond. $1.44363$
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  + (−0.400 + 0.193i)2-s + (−1.12 + 1.40i)4-s + (0.623 − 0.300i)5-s + (0.222 + 0.279i)7-s + (0.376 − 1.64i)8-s + (−0.192 + 0.240i)10-s + (1.09 + 4.81i)11-s + (1.25 + 5.51i)13-s + (−0.143 − 0.0689i)14-s + (−0.634 − 2.77i)16-s − 4.49·17-s + (−1.46 + 1.84i)19-s + (−0.277 + 1.21i)20-s + (−1.37 − 1.71i)22-s + (2.06 + 0.996i)23-s + ⋯
L(s)  = 1  + (−0.283 + 0.136i)2-s + (−0.561 + 0.704i)4-s + (0.278 − 0.134i)5-s + (0.0841 + 0.105i)7-s + (0.133 − 0.583i)8-s + (−0.0607 + 0.0761i)10-s + (0.331 + 1.45i)11-s + (0.348 + 1.52i)13-s + (−0.0382 − 0.0184i)14-s + (−0.158 − 0.694i)16-s − 1.08·17-s + (−0.337 + 0.422i)19-s + (−0.0620 + 0.271i)20-s + (−0.292 − 0.366i)22-s + (0.431 + 0.207i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00819 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00819 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.00819 - 0.999i$
Analytic conductor: \(2.08409\)
Root analytic conductor: \(1.44363\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1/2),\ -0.00819 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.657507 + 0.662916i\)
\(L(\frac12)\) \(\approx\) \(0.657507 + 0.662916i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 + (-5.09 + 1.73i)T \)
good2 \( 1 + (0.400 - 0.193i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (-0.623 + 0.300i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (-0.222 - 0.279i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (-1.09 - 4.81i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-1.25 - 5.51i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + 4.49T + 17T^{2} \)
19 \( 1 + (1.46 - 1.84i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (-2.06 - 0.996i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (-6.02 + 2.90i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + (-1.09 + 4.81i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + 3.10T + 41T^{2} \)
43 \( 1 + (-3.06 - 1.47i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (1.43 + 6.28i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (4.22 - 2.03i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 + (1.02 + 1.28i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (-0.516 + 2.26i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (1.63 + 7.15i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (5.06 + 2.44i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (-1.03 + 4.54i)T + (-71.1 - 34.2i)T^{2} \)
83 \( 1 + (2.77 - 3.48i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (5.11 - 2.46i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (0.112 - 0.141i)T + (-21.5 - 94.5i)T^{2} \)
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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

−12.18981877077635850714168889183, −11.47670789530389594505558684090, −9.993499837227625866376478768913, −9.274396270100792389846176942888, −8.521738200710267242813604037975, −7.29510214855580443732065438614, −6.49150544084476226042676691222, −4.73636897084018060157308325560, −3.99891439602125323389929509119, −2.04117326981631198805008142054, 0.859816654778507372021084367842, 2.88057983424433482653096853261, 4.51287283519280950009099470800, 5.70918621958520558953638301112, 6.50832279272118702439949090057, 8.299982742315712771413829998790, 8.705785686045249354997605614982, 9.989528072074649949647045085402, 10.69362979507544112564446423778, 11.40616979264870596233025176810

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