Properties

Degree $2$
Conductor $261$
Sign $-0.0857 + 0.996i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.170 − 1.51i)2-s + (1.63 + 0.373i)4-s + (5.91 − 4.71i)5-s + (−1.42 − 6.25i)7-s + (2.85 − 8.16i)8-s + (−6.13 − 9.75i)10-s + (5.33 + 15.2i)11-s + (−5.47 + 11.3i)13-s + (−9.72 + 1.09i)14-s + (−5.83 − 2.80i)16-s + (−11.5 − 11.5i)17-s + (12.0 − 7.60i)19-s + (11.4 − 5.50i)20-s + (23.9 − 5.47i)22-s + (2.84 − 3.56i)23-s + ⋯
L(s)  = 1  + (0.0853 − 0.757i)2-s + (0.409 + 0.0933i)4-s + (1.18 − 0.943i)5-s + (−0.204 − 0.894i)7-s + (0.357 − 1.02i)8-s + (−0.613 − 0.975i)10-s + (0.484 + 1.38i)11-s + (−0.420 + 0.874i)13-s + (−0.694 + 0.0782i)14-s + (−0.364 − 0.175i)16-s + (−0.679 − 0.679i)17-s + (0.636 − 0.400i)19-s + (0.571 − 0.275i)20-s + (1.09 − 0.248i)22-s + (0.123 − 0.155i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0857 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0857 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.0857 + 0.996i$
Motivic weight: \(2\)
Character: $\chi_{261} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ -0.0857 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.55015 - 1.68934i\)
\(L(\frac12)\) \(\approx\) \(1.55015 - 1.68934i\)
\(L(2)\) 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 \( 1 + (23.5 - 16.9i)T \)
good2 \( 1 + (-0.170 + 1.51i)T + (-3.89 - 0.890i)T^{2} \)
5 \( 1 + (-5.91 + 4.71i)T + (5.56 - 24.3i)T^{2} \)
7 \( 1 + (1.42 + 6.25i)T + (-44.1 + 21.2i)T^{2} \)
11 \( 1 + (-5.33 - 15.2i)T + (-94.6 + 75.4i)T^{2} \)
13 \( 1 + (5.47 - 11.3i)T + (-105. - 132. i)T^{2} \)
17 \( 1 + (11.5 + 11.5i)T + 289iT^{2} \)
19 \( 1 + (-12.0 + 7.60i)T + (156. - 325. i)T^{2} \)
23 \( 1 + (-2.84 + 3.56i)T + (-117. - 515. i)T^{2} \)
31 \( 1 + (1.32 - 11.7i)T + (-936. - 213. i)T^{2} \)
37 \( 1 + (-2.68 + 7.68i)T + (-1.07e3 - 853. i)T^{2} \)
41 \( 1 + (10.8 - 10.8i)T - 1.68e3iT^{2} \)
43 \( 1 + (41.8 - 4.71i)T + (1.80e3 - 411. i)T^{2} \)
47 \( 1 + (-28.2 + 9.87i)T + (1.72e3 - 1.37e3i)T^{2} \)
53 \( 1 + (-25.1 - 31.5i)T + (-625. + 2.73e3i)T^{2} \)
59 \( 1 - 72.0T + 3.48e3T^{2} \)
61 \( 1 + (36.0 - 57.3i)T + (-1.61e3 - 3.35e3i)T^{2} \)
67 \( 1 + (40.9 + 84.9i)T + (-2.79e3 + 3.50e3i)T^{2} \)
71 \( 1 + (13.8 - 28.7i)T + (-3.14e3 - 3.94e3i)T^{2} \)
73 \( 1 + (-14.9 - 132. i)T + (-5.19e3 + 1.18e3i)T^{2} \)
79 \( 1 + (-82.9 - 29.0i)T + (4.87e3 + 3.89e3i)T^{2} \)
83 \( 1 + (-14.4 + 63.2i)T + (-6.20e3 - 2.98e3i)T^{2} \)
89 \( 1 + (10.8 - 95.9i)T + (-7.72e3 - 1.76e3i)T^{2} \)
97 \( 1 + (-63.6 - 101. i)T + (-4.08e3 + 8.47e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.63564541323999166811102829744, −10.47319731914838077185280499481, −9.654588098137802397571609121319, −9.141935244806395129392826225819, −7.24961814606494300759480569777, −6.71565446585541040258975795904, −5.06134636705074471382661855169, −4.08577075116915945743733593310, −2.31614795487563453361866746608, −1.29940925326467205648775386116, 2.08397062210145113815694652795, 3.17002010399630653807867261971, 5.53722532846640604813304890775, 5.92973542064958383722858847411, 6.75006742463407694786207914490, 7.986164359041299141529953889985, 9.046050480442617356567354251598, 10.16500939553891067115901621523, 10.99041409607676233113892751144, 11.85441728035674249226150255876

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