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Properties

Label 2-117-117.4-c1-0-0
Degree $2$
Conductor $117$
Sign $-0.962 + 0.270i$
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  + 2.65i·2-s + (−1.38 + 1.04i)3-s − 5.03·4-s + (2.43 + 1.40i)5-s + (−2.75 − 3.67i)6-s + (0.226 + 0.130i)7-s − 8.03i·8-s + (0.835 − 2.88i)9-s + (−3.72 + 6.44i)10-s − 1.55i·11-s + (6.96 − 5.23i)12-s + (−2.26 + 2.80i)13-s + (−0.346 + 0.599i)14-s + (−4.82 + 0.585i)15-s + 11.2·16-s + (3.65 + 6.32i)17-s + ⋯
L(s)  = 1  + 1.87i·2-s + (−0.799 + 0.600i)3-s − 2.51·4-s + (1.08 + 0.627i)5-s + (−1.12 − 1.49i)6-s + (0.0854 + 0.0493i)7-s − 2.84i·8-s + (0.278 − 0.960i)9-s + (−1.17 + 2.03i)10-s − 0.469i·11-s + (2.01 − 1.51i)12-s + (−0.627 + 0.778i)13-s + (−0.0925 + 0.160i)14-s + (−1.24 + 0.151i)15-s + 2.81·16-s + (0.886 + 1.53i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.962 + 0.270i$
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: \((2,\ 117,\ (\ :1/2),\ -0.962 + 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.111373 - 0.808364i\)
\(L(\frac12)\) \(\approx\) \(0.111373 - 0.808364i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.38 - 1.04i)T \)
13 \( 1 + (2.26 - 2.80i)T \)
good2 \( 1 - 2.65iT - 2T^{2} \)
5 \( 1 + (-2.43 - 1.40i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.226 - 0.130i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.55iT - 11T^{2} \)
17 \( 1 + (-3.65 - 6.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.447 + 0.258i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.37 + 2.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.56T + 29T^{2} \)
31 \( 1 + (3.17 + 1.83i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.90 - 1.10i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.42 + 3.71i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.74 + 3.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.10 + 4.10i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.68T + 53T^{2} \)
59 \( 1 - 11.1iT - 59T^{2} \)
61 \( 1 + (-1.80 + 3.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.11 - 4.10i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.40 - 3.12i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.76iT - 73T^{2} \)
79 \( 1 + (1.36 + 2.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.18 + 4.14i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.18 + 0.684i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.7 + 6.77i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.51362429523772048384719542045, −13.56332905850775037703265215903, −12.28469940121873969968283486532, −10.52278769396617102243826353345, −9.737816529181713232838152166899, −8.664987530081538071551001182522, −7.14988892238823281369758416739, −6.12296227260965411711741813167, −5.59221088617982330569437892578, −4.15937427963732653927557890407, 1.13567168702656454179586533610, 2.58220704921345894622547146909, 4.78874136017746451913474159658, 5.58083263574051010258528612090, 7.70173061368784693882022586212, 9.376080698740547740213872188451, 9.964495407758619917620503187903, 11.01094842051569611858316109219, 12.10164108687422633225367564818, 12.62082602951884039557587595800

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