Properties

Label 2-117-117.50-c1-0-8
Degree $2$
Conductor $117$
Sign $0.857 + 0.514i$
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  + (−0.137 − 0.513i)2-s + (1.52 + 0.827i)3-s + (1.48 − 0.858i)4-s + (−0.764 − 2.85i)5-s + (0.215 − 0.894i)6-s + (−2.81 + 2.81i)7-s + (−1.39 − 1.39i)8-s + (1.63 + 2.51i)9-s + (−1.35 + 0.784i)10-s + (1.93 − 0.519i)11-s + (2.97 − 0.0757i)12-s + (−2.59 + 2.50i)13-s + (1.83 + 1.05i)14-s + (1.19 − 4.97i)15-s + (1.19 − 2.06i)16-s + (−2.38 + 4.12i)17-s + ⋯
L(s)  = 1  + (−0.0972 − 0.362i)2-s + (0.878 + 0.477i)3-s + (0.743 − 0.429i)4-s + (−0.341 − 1.27i)5-s + (0.0879 − 0.365i)6-s + (−1.06 + 1.06i)7-s + (−0.493 − 0.493i)8-s + (0.543 + 0.839i)9-s + (−0.429 + 0.248i)10-s + (0.584 − 0.156i)11-s + (0.858 − 0.0218i)12-s + (−0.720 + 0.693i)13-s + (0.490 + 0.283i)14-s + (0.309 − 1.28i)15-s + (0.298 − 0.516i)16-s + (−0.577 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.857 + 0.514i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.857 + 0.514i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24642 - 0.344964i\)
\(L(\frac12)\) \(\approx\) \(1.24642 - 0.344964i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.52 - 0.827i)T \)
13 \( 1 + (2.59 - 2.50i)T \)
good2 \( 1 + (0.137 + 0.513i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.764 + 2.85i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (2.81 - 2.81i)T - 7iT^{2} \)
11 \( 1 + (-1.93 + 0.519i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.38 - 4.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.57 + 0.958i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 0.938T + 23T^{2} \)
29 \( 1 + (3.82 + 2.20i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.0653 - 0.0175i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.00215 + 0.000576i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (5.69 - 5.69i)T - 41iT^{2} \)
43 \( 1 + 9.98iT - 43T^{2} \)
47 \( 1 + (-1.52 + 5.69i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + 7.99iT - 53T^{2} \)
59 \( 1 + (-1.12 + 4.21i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 + (0.773 + 0.773i)T + 67iT^{2} \)
71 \( 1 + (-4.06 - 15.1i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.854 + 0.854i)T - 73iT^{2} \)
79 \( 1 + (0.501 + 0.868i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.70 - 2.06i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (0.189 - 0.708i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.74 - 3.74i)T + 97iT^{2} \)
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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

−13.25012989915545562176968187316, −12.35673745253753918891535642911, −11.55509384819544424753852861016, −9.943954059661916152486204269100, −9.287231929425512324880797742498, −8.470150905628942957713529519930, −6.80677047820002188970576342453, −5.33941681845534210603754722541, −3.71893607440976739381802980234, −2.13410334660338962379432314476, 2.78913912683301661534737580827, 3.59387305837894436687106784115, 6.46982878878285917350396718887, 7.18546716779194736897495606275, 7.65025555328892502316003229937, 9.367052639282205563556006733812, 10.42412310213625408275124041825, 11.61345724726394065071719712833, 12.68944481002571559799705985905, 13.80096114436670297556191737339

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