Properties

Degree $2$
Conductor $261$
Sign $-0.968 - 0.248i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.887 − 2.53i)2-s + (−2.51 − 2.00i)4-s + (−3.04 − 6.31i)5-s + (−0.484 − 0.608i)7-s + (1.77 − 1.11i)8-s + (−18.7 + 2.10i)10-s + (−4.24 − 2.66i)11-s + (−2.96 + 0.675i)13-s + (−1.97 + 0.690i)14-s + (−4.11 − 18.0i)16-s + (−5.44 + 5.44i)17-s + (1.87 + 16.6i)19-s + (−5.02 + 22.0i)20-s + (−10.5 + 8.40i)22-s + (−16.4 − 7.94i)23-s + ⋯
L(s)  = 1  + (0.443 − 1.26i)2-s + (−0.629 − 0.502i)4-s + (−0.608 − 1.26i)5-s + (−0.0692 − 0.0868i)7-s + (0.221 − 0.139i)8-s + (−1.87 + 0.210i)10-s + (−0.386 − 0.242i)11-s + (−0.227 + 0.0519i)13-s + (−0.140 + 0.0493i)14-s + (−0.257 − 1.12i)16-s + (−0.320 + 0.320i)17-s + (0.0988 + 0.877i)19-s + (−0.251 + 1.10i)20-s + (−0.479 + 0.381i)22-s + (−0.717 − 0.345i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.968 - 0.248i$
Motivic weight: \(2\)
Character: $\chi_{261} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ -0.968 - 0.248i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.186874 + 1.48056i\)
\(L(\frac12)\) \(\approx\) \(0.186874 + 1.48056i\)
\(L(2)\) 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 \( 1 + (10.4 + 27.0i)T \)
good2 \( 1 + (-0.887 + 2.53i)T + (-3.12 - 2.49i)T^{2} \)
5 \( 1 + (3.04 + 6.31i)T + (-15.5 + 19.5i)T^{2} \)
7 \( 1 + (0.484 + 0.608i)T + (-10.9 + 47.7i)T^{2} \)
11 \( 1 + (4.24 + 2.66i)T + (52.4 + 109. i)T^{2} \)
13 \( 1 + (2.96 - 0.675i)T + (152. - 73.3i)T^{2} \)
17 \( 1 + (5.44 - 5.44i)T - 289iT^{2} \)
19 \( 1 + (-1.87 - 16.6i)T + (-351. + 80.3i)T^{2} \)
23 \( 1 + (16.4 + 7.94i)T + (329. + 413. i)T^{2} \)
31 \( 1 + (-7.35 + 21.0i)T + (-751. - 599. i)T^{2} \)
37 \( 1 + (-6.78 + 4.26i)T + (593. - 1.23e3i)T^{2} \)
41 \( 1 + (-35.8 - 35.8i)T + 1.68e3iT^{2} \)
43 \( 1 + (-61.1 + 21.3i)T + (1.44e3 - 1.15e3i)T^{2} \)
47 \( 1 + (-23.5 + 37.5i)T + (-958. - 1.99e3i)T^{2} \)
53 \( 1 + (21.6 - 10.4i)T + (1.75e3 - 2.19e3i)T^{2} \)
59 \( 1 - 90.6T + 3.48e3T^{2} \)
61 \( 1 + (-10.2 - 1.15i)T + (3.62e3 + 828. i)T^{2} \)
67 \( 1 + (-20.7 - 4.72i)T + (4.04e3 + 1.94e3i)T^{2} \)
71 \( 1 + (88.7 - 20.2i)T + (4.54e3 - 2.18e3i)T^{2} \)
73 \( 1 + (-2.59 - 7.42i)T + (-4.16e3 + 3.32e3i)T^{2} \)
79 \( 1 + (-13.3 - 21.2i)T + (-2.70e3 + 5.62e3i)T^{2} \)
83 \( 1 + (-81.3 + 101. i)T + (-1.53e3 - 6.71e3i)T^{2} \)
89 \( 1 + (-34.1 + 97.6i)T + (-6.19e3 - 4.93e3i)T^{2} \)
97 \( 1 + (-113. + 12.7i)T + (9.17e3 - 2.09e3i)T^{2} \)
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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

−11.53919199054332482654926642061, −10.46894469195366854363377503029, −9.588224098636271555490718484477, −8.435513141879496893647728140152, −7.54709413028619407501258422986, −5.76986413225482683446117816651, −4.50957515071852113173125384397, −3.82393112979905085663628819143, −2.21682406662470943153775197556, −0.66181131167531333376279100581, 2.65865679669143151586571803859, 4.11582088644812397608246214106, 5.33059666340175459009897915403, 6.47613103445682204004185092271, 7.24104581407421809967205119256, 7.83937103374293016153619165581, 9.184192169234621471041592424290, 10.59600467046731985248662081671, 11.16853416798699623080871543178, 12.40511572941677410662932513547

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