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Properties

Label 2-117-117.110-c1-0-7
Degree $2$
Conductor $117$
Sign $0.459 - 0.888i$
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  + (−0.477 + 1.78i)2-s + (1.66 − 0.488i)3-s + (−1.21 − 0.699i)4-s + (0.672 − 2.51i)5-s + (0.0772 + 3.19i)6-s + (1.99 + 1.99i)7-s + (−0.782 + 0.782i)8-s + (2.52 − 1.62i)9-s + (4.14 + 2.39i)10-s + (−6.08 − 1.62i)11-s + (−2.35 − 0.570i)12-s + (−3.52 + 0.756i)13-s + (−4.50 + 2.59i)14-s + (−0.108 − 4.49i)15-s + (−2.42 − 4.19i)16-s + (2.31 + 4.01i)17-s + ⋯
L(s)  = 1  + (−0.337 + 1.25i)2-s + (0.959 − 0.282i)3-s + (−0.606 − 0.349i)4-s + (0.300 − 1.12i)5-s + (0.0315 + 1.30i)6-s + (0.753 + 0.753i)7-s + (−0.276 + 0.276i)8-s + (0.840 − 0.541i)9-s + (1.31 + 0.757i)10-s + (−1.83 − 0.491i)11-s + (−0.680 − 0.164i)12-s + (−0.977 + 0.209i)13-s + (−1.20 + 0.694i)14-s + (−0.0281 − 1.16i)15-s + (−0.605 − 1.04i)16-s + (0.562 + 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.459 - 0.888i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.459 - 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02638 + 0.624876i\)
\(L(\frac12)\) \(\approx\) \(1.02638 + 0.624876i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.66 + 0.488i)T \)
13 \( 1 + (3.52 - 0.756i)T \)
good2 \( 1 + (0.477 - 1.78i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-0.672 + 2.51i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.99 - 1.99i)T + 7iT^{2} \)
11 \( 1 + (6.08 + 1.62i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.31 - 4.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.390 - 0.104i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 4.26T + 23T^{2} \)
29 \( 1 + (-5.22 + 3.01i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.27 + 0.608i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.60 + 0.431i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.18 + 1.18i)T + 41iT^{2} \)
43 \( 1 + 2.99iT - 43T^{2} \)
47 \( 1 + (-1.25 - 4.68i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 0.186iT - 53T^{2} \)
59 \( 1 + (-1.07 - 4.01i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 0.0950T + 61T^{2} \)
67 \( 1 + (2.70 - 2.70i)T - 67iT^{2} \)
71 \( 1 + (1.56 - 5.83i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (1.49 + 1.49i)T + 73iT^{2} \)
79 \( 1 + (2.84 - 4.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-15.0 + 4.04i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.0313 - 0.117i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-11.2 + 11.2i)T - 97iT^{2} \)
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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

−13.97447094181510893002927663927, −12.89209379482241208968850921777, −12.02611285491693032535825707847, −10.08880458476498534487043353912, −8.870463271698318399706006797246, −8.202421230046251625363652583533, −7.61838820977838363017697997802, −5.84678227089217171406201953801, −4.91616114284997575797896804904, −2.38504875405176328094079157285, 2.27386615464138490997597537254, 3.10504066858317787569541710558, 4.82196585490698168031221652663, 7.17384873068452999345290588036, 7.991105910689738883208607629240, 9.678255390555865357407578879349, 10.31702827519253926387010627945, 10.80669523991728407454494813274, 12.20954780413521555030471225557, 13.36427386502182742795954231135

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