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Properties

Label 2-117-39.5-c1-0-1
Degree $2$
Conductor $117$
Sign $0.0956 - 0.995i$
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  + (1.77 + 1.77i)2-s + 4.32i·4-s + (−2.34 − 2.34i)5-s + (1.51 + 1.51i)7-s + (−4.12 + 4.12i)8-s − 8.34i·10-s + (2.34 − 2.34i)11-s + (−0.806 − 3.51i)13-s + 5.38i·14-s − 6.02·16-s − 4.24·17-s + (−0.193 + 0.193i)19-s + (10.1 − 10.1i)20-s + 8.34·22-s − 2.41·23-s + ⋯
L(s)  = 1  + (1.25 + 1.25i)2-s + 2.16i·4-s + (−1.05 − 1.05i)5-s + (0.572 + 0.572i)7-s + (−1.45 + 1.45i)8-s − 2.64i·10-s + (0.708 − 0.708i)11-s + (−0.223 − 0.974i)13-s + 1.43i·14-s − 1.50·16-s − 1.02·17-s + (−0.0443 + 0.0443i)19-s + (2.26 − 2.26i)20-s + 1.78·22-s − 0.503·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0956 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0956 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.0956 - 0.995i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.0956 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25076 + 1.13631i\)
\(L(\frac12)\) \(\approx\) \(1.25076 + 1.13631i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (0.806 + 3.51i)T \)
good2 \( 1 + (-1.77 - 1.77i)T + 2iT^{2} \)
5 \( 1 + (2.34 + 2.34i)T + 5iT^{2} \)
7 \( 1 + (-1.51 - 1.51i)T + 7iT^{2} \)
11 \( 1 + (-2.34 + 2.34i)T - 11iT^{2} \)
17 \( 1 + 4.24T + 17T^{2} \)
19 \( 1 + (0.193 - 0.193i)T - 19iT^{2} \)
23 \( 1 + 2.41T + 23T^{2} \)
29 \( 1 - 7.56iT - 29T^{2} \)
31 \( 1 + (4.83 - 4.83i)T - 31iT^{2} \)
37 \( 1 + (-1 - i)T + 37iT^{2} \)
41 \( 1 + (-1.89 - 1.89i)T + 41iT^{2} \)
43 \( 1 + 5.67iT - 43T^{2} \)
47 \( 1 + (-2.34 + 2.34i)T - 47iT^{2} \)
53 \( 1 - 1.96iT - 53T^{2} \)
59 \( 1 + (-0.753 + 0.753i)T - 59iT^{2} \)
61 \( 1 + 1.61T + 61T^{2} \)
67 \( 1 + (-7.51 + 7.51i)T - 67iT^{2} \)
71 \( 1 + (-1.66 - 1.66i)T + 71iT^{2} \)
73 \( 1 + (-7.34 - 7.34i)T + 73iT^{2} \)
79 \( 1 + 16.6T + 79T^{2} \)
83 \( 1 + (-7.73 - 7.73i)T + 83iT^{2} \)
89 \( 1 + (-4.76 + 4.76i)T - 89iT^{2} \)
97 \( 1 + (0.707 - 0.707i)T - 97iT^{2} \)
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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

−13.93699796788968839764438242151, −12.76745082168094180017675885173, −12.23254845752936808486776566298, −11.19779252978881214847178400715, −8.760536495868758716674221878505, −8.259890737347159172991266007605, −7.06487686684131482540101758301, −5.62768726226208946901834615856, −4.75259024631590336883977425553, −3.60300725588795158998113699336, 2.21858695598853181218330724490, 3.92587714666624561992388678579, 4.43209503923633659507332246862, 6.41306535681516405778801657012, 7.56279236896360386515687817850, 9.540089317259007959817320104742, 10.76511884655636013559458421095, 11.42379014269391673006619401464, 11.97348197017180557905185576456, 13.25078399051585285032424513527

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