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Properties

Label 2-117-13.11-c2-0-3
Degree $2$
Conductor $117$
Sign $0.824 - 0.565i$
Analytic cond. $3.18801$
Root an. cond. $1.78550$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.46 + 2i)4-s + (1.42 + 5.33i)7-s + (12.9 − 0.5i)13-s + (7.99 + 13.8i)16-s + (−1.16 + 0.313i)19-s − 25i·25-s + (−5.71 + 21.3i)28-s + (−36.8 − 36.8i)31-s + (−65.0 − 17.4i)37-s + (−19.5 − 11.2i)43-s + (16.0 − 9.27i)49-s + (46 + 24.2i)52-s + (56.2 − 97.5i)61-s + 63.9i·64-s + (−22.0 + 82.2i)67-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)4-s + (0.204 + 0.761i)7-s + (0.999 − 0.0384i)13-s + (0.499 + 0.866i)16-s + (−0.0615 + 0.0164i)19-s − i·25-s + (−0.204 + 0.761i)28-s + (−1.18 − 1.18i)31-s + (−1.75 − 0.471i)37-s + (−0.453 − 0.261i)43-s + (0.327 − 0.189i)49-s + (0.884 + 0.466i)52-s + (0.922 − 1.59i)61-s + 0.999i·64-s + (−0.328 + 1.22i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.824 - 0.565i$
Analytic conductor: \(3.18801\)
Root analytic conductor: \(1.78550\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1),\ 0.824 - 0.565i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.59097 + 0.492740i\)
\(L(\frac12)\) \(\approx\) \(1.59097 + 0.492740i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-12.9 + 0.5i)T \)
good2 \( 1 + (-3.46 - 2i)T^{2} \)
5 \( 1 + 25iT^{2} \)
7 \( 1 + (-1.42 - 5.33i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (104. + 60.5i)T^{2} \)
17 \( 1 + (144.5 + 250. i)T^{2} \)
19 \( 1 + (1.16 - 0.313i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (36.8 + 36.8i)T + 961iT^{2} \)
37 \( 1 + (65.0 + 17.4i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (19.5 + 11.2i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 2.20e3iT^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-56.2 + 97.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (22.0 - 82.2i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (56.7 - 56.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 157.T + 6.24e3T^{2} \)
83 \( 1 + 6.88e3iT^{2} \)
89 \( 1 + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-180. + 48.3i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−13.21783517286361333086402475544, −12.23703553393297187050193477804, −11.42282693779055579695653645625, −10.47114227840686612229981585327, −8.924161589659249322478223566610, −8.006089256533648792999086117659, −6.71220808728986516728608506087, −5.60427385338968343450665617269, −3.69103721176302777319168334967, −2.12990784565480443851830159081, 1.49233516272456680939190834537, 3.49689878434552070953111816313, 5.27431052887705394430799525353, 6.57800982786059469241387327744, 7.51616240557551373361615703374, 8.938081568196808006647498396745, 10.37128567949318572105452408702, 10.95152722435445926746450315856, 11.99196413926986638677819399841, 13.30766850756355281974724728063

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