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Properties

Label 2-117-117.110-c1-0-3
Degree $2$
Conductor $117$
Sign $0.925 - 0.378i$
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  + (0.308 − 1.14i)2-s + (−0.781 + 1.54i)3-s + (0.505 + 0.291i)4-s + (−0.744 + 2.77i)5-s + (1.53 + 1.37i)6-s + (1.04 + 1.04i)7-s + (2.17 − 2.17i)8-s + (−1.77 − 2.41i)9-s + (2.96 + 1.71i)10-s + (−1.68 − 0.451i)11-s + (−0.845 + 0.552i)12-s + (1.90 − 3.06i)13-s + (1.52 − 0.878i)14-s + (−3.71 − 3.32i)15-s + (−1.24 − 2.15i)16-s + (−0.939 − 1.62i)17-s + ⋯
L(s)  = 1  + (0.217 − 0.812i)2-s + (−0.451 + 0.892i)3-s + (0.252 + 0.145i)4-s + (−0.332 + 1.24i)5-s + (0.627 + 0.561i)6-s + (0.394 + 0.394i)7-s + (0.768 − 0.768i)8-s + (−0.592 − 0.805i)9-s + (0.937 + 0.541i)10-s + (−0.507 − 0.135i)11-s + (−0.244 + 0.159i)12-s + (0.527 − 0.849i)13-s + (0.406 − 0.234i)14-s + (−0.957 − 0.857i)15-s + (−0.311 − 0.539i)16-s + (−0.227 − 0.394i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.925 - 0.378i$
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: \((2,\ 117,\ (\ :1/2),\ 0.925 - 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09688 + 0.215759i\)
\(L(\frac12)\) \(\approx\) \(1.09688 + 0.215759i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.781 - 1.54i)T \)
13 \( 1 + (-1.90 + 3.06i)T \)
good2 \( 1 + (-0.308 + 1.14i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (0.744 - 2.77i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.04 - 1.04i)T + 7iT^{2} \)
11 \( 1 + (1.68 + 0.451i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.939 + 1.62i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.34 - 1.16i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 + (-5.80 + 3.35i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.52 + 0.676i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (11.1 - 2.99i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.459 + 0.459i)T + 41iT^{2} \)
43 \( 1 + 2.05iT - 43T^{2} \)
47 \( 1 + (-0.619 - 2.31i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 + (-3.35 - 12.5i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 + (2.56 - 2.56i)T - 67iT^{2} \)
71 \( 1 + (2.99 - 11.1i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (8.97 + 8.97i)T + 73iT^{2} \)
79 \( 1 + (4.85 - 8.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.65 + 0.712i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-1.57 - 5.86i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.70 - 3.70i)T - 97iT^{2} \)
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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

−13.55484175489777320379799850576, −12.01898133473597276202504928897, −11.51848403173319676401782832475, −10.54784040617865059944712890342, −10.06486053792305296304102682018, −8.234134133256099712781454945978, −6.89188589250066033594518386758, −5.46870531504823349437243387752, −3.78719279101234570975681958900, −2.80428376399100307049327467098, 1.57862011263618646345486119251, 4.63959374659302251712745606530, 5.60261133114989954028892129197, 6.84877405427248757668618768710, 7.82539004396427673007392373552, 8.686696366569512549471305847039, 10.58646429831134632018564719333, 11.62942653360101771506032683122, 12.47464346583277953415485265527, 13.64107517795989868032080553058

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