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Properties

Label 2-117-117.11-c1-0-1
Degree $2$
Conductor $117$
Sign $0.378 - 0.925i$
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  + (−1.75 + 1.75i)2-s + (−1.38 − 1.03i)3-s − 4.17i·4-s + (3.52 + 0.943i)5-s + (4.25 − 0.617i)6-s + (−1.05 − 0.281i)7-s + (3.81 + 3.81i)8-s + (0.852 + 2.87i)9-s + (−7.84 + 4.52i)10-s + (1.46 + 1.46i)11-s + (−4.32 + 5.78i)12-s + (3.60 + 0.0594i)13-s + (2.34 − 1.35i)14-s + (−3.90 − 4.95i)15-s − 5.05·16-s + (−2.40 + 4.17i)17-s + ⋯
L(s)  = 1  + (−1.24 + 1.24i)2-s + (−0.801 − 0.598i)3-s − 2.08i·4-s + (1.57 + 0.421i)5-s + (1.73 − 0.252i)6-s + (−0.397 − 0.106i)7-s + (1.34 + 1.34i)8-s + (0.284 + 0.958i)9-s + (−2.47 + 1.43i)10-s + (0.442 + 0.442i)11-s + (−1.24 + 1.67i)12-s + (0.999 + 0.0164i)13-s + (0.626 − 0.361i)14-s + (−1.00 − 1.27i)15-s − 1.26·16-s + (−0.584 + 1.01i)17-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.481460 + 0.323407i\)
\(L(\frac12)\) \(\approx\) \(0.481460 + 0.323407i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.38 + 1.03i)T \)
13 \( 1 + (-3.60 - 0.0594i)T \)
good2 \( 1 + (1.75 - 1.75i)T - 2iT^{2} \)
5 \( 1 + (-3.52 - 0.943i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.05 + 0.281i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.46 - 1.46i)T + 11iT^{2} \)
17 \( 1 + (2.40 - 4.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.01 + 1.34i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.496 + 0.860i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.59iT - 29T^{2} \)
31 \( 1 + (-0.832 + 3.10i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (3.11 + 0.834i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.0289 + 0.108i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (7.98 - 4.61i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.57 - 2.02i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + 9.48iT - 53T^{2} \)
59 \( 1 + (-0.399 - 0.399i)T + 59iT^{2} \)
61 \( 1 + (-4.06 - 7.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.34 + 1.43i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.773 + 2.88i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-4.89 + 4.89i)T - 73iT^{2} \)
79 \( 1 + (5.66 - 9.80i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.638 - 2.38i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-0.0437 + 0.163i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.80 + 6.75i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.78906149659724773919716147713, −13.04866536851465901168757940187, −11.25070930917446704697419848966, −10.17834483972844572378804888920, −9.534241322988936873619259183412, −8.233595029901006037449580233618, −6.76752142275052530490703079959, −6.38457167427000061768185465682, −5.42050508945218294715953606913, −1.59609120152007447449190606302, 1.31935030372453493567548881692, 3.28227280892103264312737790922, 5.31176400200569388108379584681, 6.61140149039016115418631610230, 8.745962698586330733749900804166, 9.421892172865802395803539726328, 10.07254066440050213481032819139, 11.04504552501214485179387906788, 11.90514079846360584518877703436, 12.99477532183159590566257946351

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