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Properties

Label 2-117-117.4-c1-0-11
Degree $2$
Conductor $117$
Sign $-0.999 - 0.0149i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
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  − 2.31i·2-s + (−0.644 − 1.60i)3-s − 3.36·4-s + (−1.09 − 0.632i)5-s + (−3.72 + 1.49i)6-s + (3.61 + 2.08i)7-s + 3.16i·8-s + (−2.16 + 2.07i)9-s + (−1.46 + 2.53i)10-s − 2.71i·11-s + (2.16 + 5.41i)12-s + (1.54 − 3.26i)13-s + (4.83 − 8.37i)14-s + (−0.310 + 2.16i)15-s + 0.601·16-s + (1.05 + 1.82i)17-s + ⋯
L(s)  = 1  − 1.63i·2-s + (−0.372 − 0.928i)3-s − 1.68·4-s + (−0.489 − 0.282i)5-s + (−1.52 + 0.609i)6-s + (1.36 + 0.789i)7-s + 1.11i·8-s + (−0.723 + 0.690i)9-s + (−0.463 + 0.802i)10-s − 0.818i·11-s + (0.626 + 1.56i)12-s + (0.427 − 0.904i)13-s + (1.29 − 2.23i)14-s + (−0.0802 + 0.559i)15-s + 0.150·16-s + (0.254 + 0.441i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.999 - 0.0149i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.999 - 0.0149i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00662664 + 0.883606i\)
\(L(\frac12)\) \(\approx\) \(0.00662664 + 0.883606i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.644 + 1.60i)T \)
13 \( 1 + (-1.54 + 3.26i)T \)
good2 \( 1 + 2.31iT - 2T^{2} \)
5 \( 1 + (1.09 + 0.632i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-3.61 - 2.08i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.71iT - 11T^{2} \)
17 \( 1 + (-1.05 - 1.82i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.74 - 1.58i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.226 + 0.391i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.58T + 29T^{2} \)
31 \( 1 + (-3.34 - 1.92i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.55 - 4.35i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.17 + 1.25i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.49 + 4.31i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (10.8 - 6.24i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.14T + 53T^{2} \)
59 \( 1 - 6.32iT - 59T^{2} \)
61 \( 1 + (-2.54 + 4.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.2 - 7.09i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.49 - 3.75i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.95iT - 73T^{2} \)
79 \( 1 + (1.82 + 3.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.96 + 1.13i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.31 - 0.760i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.06 + 1.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.58212561471687744461942560246, −11.91845996586869925383458754761, −11.24865358182537752933897029160, −10.40150649858480662355420927942, −8.460521335206128381639365358571, −8.195296685255520533688771420804, −5.96022549433842330541598074468, −4.59162535583753159383390221200, −2.74246350437818505248268494390, −1.20623576046263260766595508286, 4.29053832946573329179338460929, 4.82038622522041634169624524074, 6.36164484708095521267123448442, 7.45511270108415439216659423093, 8.389513121593404533987635729622, 9.593854062827954372035675422852, 10.94183237937077044390745836015, 11.73742801187430811426567293488, 13.63979002690233753446100325544, 14.57778113836017357684851986551

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