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Properties

Label 2-117-117.11-c1-0-7
Degree $2$
Conductor $117$
Sign $0.955 + 0.294i$
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  + (1.30 − 1.30i)2-s + (−0.407 + 1.68i)3-s − 1.39i·4-s + (2.51 + 0.672i)5-s + (1.66 + 2.72i)6-s + (−2.72 − 0.729i)7-s + (0.782 + 0.782i)8-s + (−2.66 − 1.37i)9-s + (4.14 − 2.39i)10-s + (−4.45 − 4.45i)11-s + (2.35 + 0.570i)12-s + (2.41 − 2.67i)13-s + (−4.50 + 2.59i)14-s + (−2.15 + 3.95i)15-s + 4.84·16-s + (−2.31 + 4.01i)17-s + ⋯
L(s)  = 1  + (0.921 − 0.921i)2-s + (−0.235 + 0.971i)3-s − 0.699i·4-s + (1.12 + 0.300i)5-s + (0.679 + 1.11i)6-s + (−1.02 − 0.275i)7-s + (0.276 + 0.276i)8-s + (−0.889 − 0.457i)9-s + (1.31 − 0.757i)10-s + (−1.34 − 1.34i)11-s + (0.680 + 0.164i)12-s + (0.670 − 0.741i)13-s + (−1.20 + 0.694i)14-s + (−0.556 + 1.02i)15-s + 1.21·16-s + (−0.562 + 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.955 + 0.294i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.955 + 0.294i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53983 - 0.231827i\)
\(L(\frac12)\) \(\approx\) \(1.53983 - 0.231827i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.407 - 1.68i)T \)
13 \( 1 + (-2.41 + 2.67i)T \)
good2 \( 1 + (-1.30 + 1.30i)T - 2iT^{2} \)
5 \( 1 + (-2.51 - 0.672i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (2.72 + 0.729i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (4.45 + 4.45i)T + 11iT^{2} \)
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 \( 1 + (2.13 - 3.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.03iT - 29T^{2} \)
31 \( 1 + (-0.608 + 2.27i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.60 - 0.431i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.432 - 1.61i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.59 + 1.49i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.68 + 1.25i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 0.186iT - 53T^{2} \)
59 \( 1 + (2.93 + 2.93i)T + 59iT^{2} \)
61 \( 1 + (0.0475 + 0.0823i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.70 + 0.991i)T + (58.0 - 33.5i)T^{2} \)
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 \( 1 + (-4.04 - 15.0i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (0.0313 - 0.117i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.10 + 15.3i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.40918383507428584332248588322, −12.72616419970948279006251766281, −11.09756083974005509581646195938, −10.60427066617331982776717805793, −9.830030381977758392590387714168, −8.365031726725104936638238111520, −6.04569148176844094118956659728, −5.45058146711666799139638224742, −3.70278948057490926884432570132, −2.81494336001925647157169053897, 2.34199664037495849804596507573, 4.78976602608832900219443797360, 5.89453037888053025232744129909, 6.59527086460626451549795698267, 7.62250063481244622413036189956, 9.282214593175046627797849423133, 10.35759372065274573237676988552, 12.13503157204824478426065642086, 13.08197047981453262932291498121, 13.39038756652923598521578910912

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