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Properties

Label 2-117-117.29-c2-0-19
Degree $2$
Conductor $117$
Sign $-0.214 + 0.976i$
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  − 2.57i·2-s + (2.93 + 0.598i)3-s − 2.65·4-s + (−0.817 + 0.471i)5-s + (1.54 − 7.58i)6-s + (−2.52 − 4.37i)7-s − 3.47i·8-s + (8.28 + 3.52i)9-s + (1.21 + 2.10i)10-s − 15.9i·11-s + (−7.79 − 1.58i)12-s + (11.8 + 5.43i)13-s + (−11.2 + 6.50i)14-s + (−2.68 + 0.897i)15-s − 19.5·16-s + (21.5 + 12.4i)17-s + ⋯
L(s)  = 1  − 1.28i·2-s + (0.979 + 0.199i)3-s − 0.662·4-s + (−0.163 + 0.0943i)5-s + (0.257 − 1.26i)6-s + (−0.360 − 0.624i)7-s − 0.434i·8-s + (0.920 + 0.391i)9-s + (0.121 + 0.210i)10-s − 1.45i·11-s + (−0.649 − 0.132i)12-s + (0.908 + 0.418i)13-s + (−0.805 + 0.464i)14-s + (−0.178 + 0.0598i)15-s − 1.22·16-s + (1.26 + 0.732i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.214 + 0.976i$
Analytic conductor: \(3.18801\)
Root analytic conductor: \(1.78550\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1),\ -0.214 + 0.976i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.14915 - 1.42907i\)
\(L(\frac12)\) \(\approx\) \(1.14915 - 1.42907i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.93 - 0.598i)T \)
13 \( 1 + (-11.8 - 5.43i)T \)
good2 \( 1 + 2.57iT - 4T^{2} \)
5 \( 1 + (0.817 - 0.471i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (2.52 + 4.37i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + 15.9iT - 121T^{2} \)
17 \( 1 + (-21.5 - 12.4i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (17.4 - 30.1i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (17.7 + 10.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 17.8iT - 841T^{2} \)
31 \( 1 + (-10.2 - 17.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-1.09 - 1.90i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-5.08 - 2.93i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (0.939 + 1.62i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (6.18 + 3.56i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 34.1iT - 2.80e3T^{2} \)
59 \( 1 - 32.6iT - 3.48e3T^{2} \)
61 \( 1 + (6.68 + 11.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-39.2 + 68.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (121. + 70.0i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + 89.5T + 5.32e3T^{2} \)
79 \( 1 + (-28.9 + 50.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-38.2 - 22.0i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-48.6 + 28.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-78.3 - 135. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.00017931659665102001034227687, −11.93850181053301176613686546606, −10.63312625098658668607168244304, −10.22327977354847626098550016611, −8.887086732153804382586827988248, −7.910778266046756332727538003262, −6.25279148600564920407249015950, −3.82305139324331312555005236993, −3.41215157119949485847327566644, −1.51137312334447032293300231862, 2.49323786490711193784124628568, 4.43691277373653046695308666499, 5.99140828075042489037993880204, 7.14958501506048020222809460016, 7.980074907005405998777122462112, 8.970065122371418594414797349128, 9.963982708939818209318173165203, 11.78528233057469270803851223190, 12.88713110991431926838116896398, 13.85219929432921353903589933378

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