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Properties

Label 2-261-29.16-c1-0-0
Degree $2$
Conductor $261$
Sign $-0.455 - 0.890i$
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  + (−0.626 − 0.301i)2-s + (−0.945 − 1.18i)4-s + (−1.81 − 0.875i)5-s + (−1.49 + 1.87i)7-s + (0.543 + 2.38i)8-s + (0.874 + 1.09i)10-s + (−0.213 + 0.933i)11-s + (−1.45 + 6.36i)13-s + (1.49 − 0.722i)14-s + (−0.297 + 1.30i)16-s + 3.81·17-s + (−2.69 − 3.37i)19-s + (0.681 + 2.98i)20-s + (0.415 − 0.520i)22-s + (−4.85 + 2.33i)23-s + ⋯
L(s)  = 1  + (−0.442 − 0.213i)2-s + (−0.472 − 0.593i)4-s + (−0.813 − 0.391i)5-s + (−0.564 + 0.707i)7-s + (0.192 + 0.842i)8-s + (0.276 + 0.346i)10-s + (−0.0642 + 0.281i)11-s + (−0.403 + 1.76i)13-s + (0.400 − 0.192i)14-s + (−0.0742 + 0.325i)16-s + 0.925·17-s + (−0.617 − 0.774i)19-s + (0.152 + 0.667i)20-s + (0.0884 − 0.110i)22-s + (−1.01 + 0.487i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.455 - 0.890i$
Analytic conductor: \(2.08409\)
Root analytic conductor: \(1.44363\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1/2),\ -0.455 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.123479 + 0.201951i\)
\(L(\frac12)\) \(\approx\) \(0.123479 + 0.201951i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.32 - 0.822i)T \)
good2 \( 1 + (0.626 + 0.301i)T + (1.24 + 1.56i)T^{2} \)
5 \( 1 + (1.81 + 0.875i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (1.49 - 1.87i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (0.213 - 0.933i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (1.45 - 6.36i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 - 3.81T + 17T^{2} \)
19 \( 1 + (2.69 + 3.37i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (4.85 - 2.33i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (2.46 + 1.18i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + (-0.414 - 1.81i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 + (-3.33 + 1.60i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (0.719 - 3.15i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (-2.04 - 0.983i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 - 9.30T + 59T^{2} \)
61 \( 1 + (-1.95 + 2.45i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + (2.69 + 11.7i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (1.02 - 4.48i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (8.19 - 3.94i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + (0.954 + 4.18i)T + (-71.1 + 34.2i)T^{2} \)
83 \( 1 + (10.0 + 12.6i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-13.9 - 6.70i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-9.82 - 12.3i)T + (-21.5 + 94.5i)T^{2} \)
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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

−11.99935427915516616712334846036, −11.52050884775073201612413946383, −10.18339999802973605416188626954, −9.343481089730315419761399388811, −8.731637813571187603982156785370, −7.56919820762153410121154207618, −6.25672324148727644390745311349, −5.03085187515611337291622211218, −3.96615399792685282032561198038, −1.99847847535610802540831109740, 0.20994719260610021409645401650, 3.28419823732642817884397142240, 3.93757402737235400987281864287, 5.61795107436313564299494748832, 7.10933492593167496894997341344, 7.79803551600857499810598413084, 8.472620813595649038977503306375, 9.988450148335649378232659379224, 10.36455833010118623965592339382, 11.76442737593160438935981671097

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