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Properties

Label 2-261-29.9-c1-0-11
Degree $2$
Conductor $261$
Sign $-0.980 + 0.198i$
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  + (−0.492 − 1.02i)2-s + (0.442 − 0.554i)4-s + (−1.55 + 0.748i)5-s + (−1.92 − 2.41i)7-s + (−3.00 − 0.684i)8-s + (1.53 + 1.22i)10-s + (−1.58 + 0.361i)11-s + (−0.279 − 1.22i)13-s + (−1.52 + 3.16i)14-s + (0.462 + 2.02i)16-s − 6.59i·17-s + (−1.15 − 0.920i)19-s + (−0.272 + 1.19i)20-s + (1.15 + 1.44i)22-s + (3.19 + 1.53i)23-s + ⋯
L(s)  = 1  + (−0.348 − 0.723i)2-s + (0.221 − 0.277i)4-s + (−0.694 + 0.334i)5-s + (−0.727 − 0.912i)7-s + (−1.06 − 0.242i)8-s + (0.484 + 0.386i)10-s + (−0.477 + 0.108i)11-s + (−0.0774 − 0.339i)13-s + (−0.406 + 0.844i)14-s + (0.115 + 0.506i)16-s − 1.59i·17-s + (−0.264 − 0.211i)19-s + (−0.0608 + 0.266i)20-s + (0.245 + 0.307i)22-s + (0.666 + 0.320i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.980 + 0.198i$
Analytic conductor: \(2.08409\)
Root analytic conductor: \(1.44363\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1/2),\ -0.980 + 0.198i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0628460 - 0.628318i\)
\(L(\frac12)\) \(\approx\) \(0.0628460 - 0.628318i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.02 + 5.28i)T \)
good2 \( 1 + (0.492 + 1.02i)T + (-1.24 + 1.56i)T^{2} \)
5 \( 1 + (1.55 - 0.748i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (1.92 + 2.41i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (1.58 - 0.361i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (0.279 + 1.22i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + 6.59iT - 17T^{2} \)
19 \( 1 + (1.15 + 0.920i)T + (4.22 + 18.5i)T^{2} \)
23 \( 1 + (-3.19 - 1.53i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (-3.45 - 7.17i)T + (-19.3 + 24.2i)T^{2} \)
37 \( 1 + (4.53 + 1.03i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 - 1.22iT - 41T^{2} \)
43 \( 1 + (-4.21 + 8.75i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (-7.72 + 1.76i)T + (42.3 - 20.3i)T^{2} \)
53 \( 1 + (-10.7 + 5.17i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 - 0.966T + 59T^{2} \)
61 \( 1 + (0.188 - 0.150i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + (-1.60 + 7.02i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (-1.34 - 5.90i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-1.97 + 4.09i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + (2.50 + 0.572i)T + (71.1 + 34.2i)T^{2} \)
83 \( 1 + (7.06 - 8.85i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (3.64 + 7.56i)T + (-55.4 + 69.5i)T^{2} \)
97 \( 1 + (-5.80 - 4.62i)T + (21.5 + 94.5i)T^{2} \)
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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

−11.42094814023729050868047146393, −10.55429977350980718442091330751, −9.936247951941058198116035564929, −8.910236988186574472827463891795, −7.40923943955843850953782238695, −6.82731615812576302936719954240, −5.32244514517027276552980380377, −3.70878894953337308563144740209, −2.67314091894042627937024832118, −0.52336414614974028247239394101, 2.63700543491302302827845633558, 4.00202515926463466120076757730, 5.68406485570586014560533172828, 6.48282200971809342961283297931, 7.66046850633808851986471106396, 8.480964083130759793604399867313, 9.142820995424128256582070218511, 10.49507593239935968372272796719, 11.67150992509495617515880666452, 12.43909448008182435623494554164

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