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Properties

Label 2-261-29.7-c1-0-7
Degree $2$
Conductor $261$
Sign $0.995 + 0.0915i$
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  + (0.277 + 1.21i)2-s + (0.400 − 0.193i)4-s + (−0.900 − 3.94i)5-s + (−0.623 − 0.300i)7-s + (1.90 + 2.38i)8-s + (4.54 − 2.19i)10-s + (1.77 − 2.22i)11-s + (0.914 − 1.14i)13-s + (0.192 − 0.841i)14-s + (−1.81 + 2.27i)16-s + 1.60·17-s + (2.42 − 1.16i)19-s + (−1.12 − 1.40i)20-s + (3.20 + 1.54i)22-s + (−1.14 + 5.02i)23-s + ⋯
L(s)  = 1  + (0.196 + 0.859i)2-s + (0.200 − 0.0965i)4-s + (−0.402 − 1.76i)5-s + (−0.235 − 0.113i)7-s + (0.672 + 0.842i)8-s + (1.43 − 0.692i)10-s + (0.535 − 0.672i)11-s + (0.253 − 0.318i)13-s + (0.0513 − 0.224i)14-s + (−0.453 + 0.569i)16-s + 0.388·17-s + (0.556 − 0.267i)19-s + (−0.251 − 0.315i)20-s + (0.682 + 0.328i)22-s + (−0.239 + 1.04i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.995 + 0.0915i$
Analytic conductor: \(2.08409\)
Root analytic conductor: \(1.44363\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1/2),\ 0.995 + 0.0915i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48960 - 0.0683201i\)
\(L(\frac12)\) \(\approx\) \(1.48960 - 0.0683201i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.71 - 3.89i)T \)
good2 \( 1 + (-0.277 - 1.21i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + (0.900 + 3.94i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (0.623 + 0.300i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (-1.77 + 2.22i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-0.914 + 1.14i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 - 1.60T + 17T^{2} \)
19 \( 1 + (-2.42 + 1.16i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (1.14 - 5.02i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (-0.434 - 1.90i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 + (-1.77 - 2.22i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 6.49T + 41T^{2} \)
43 \( 1 + (0.147 - 0.648i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-2.96 + 3.71i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-0.0108 - 0.0476i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 - 6.39T + 59T^{2} \)
61 \( 1 + (-1.17 - 0.567i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (-9.32 - 11.6i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (-1.40 + 1.76i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (1.85 - 8.11i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (6.07 + 7.61i)T + (-17.5 + 77.0i)T^{2} \)
83 \( 1 + (3.62 - 1.74i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-2.50 - 10.9i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (-4.11 + 1.98i)T + (60.4 - 75.8i)T^{2} \)
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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

−11.95183781166447141276674929779, −11.29985697255018577885723238497, −9.848604248672553907722472496845, −8.762198356310516403557764648355, −8.088549626394637640885466530502, −7.03388597216121491588813593297, −5.69599267313720494190247455659, −5.10073790179567967732308217134, −3.72460838542512286855467992908, −1.29248180393161245406229220757, 2.16519737766296934874470047424, 3.25191480597292024213424863461, 4.14005348771358314856806263417, 6.23889758902653480382694022604, 7.00108346058756091508480391751, 7.83563444136655278892784443339, 9.620849059016286641742721970711, 10.31629167616113078941122551031, 11.18258265108656135165304697507, 11.77519361279399896462269039936

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