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Properties

Label 2-117-117.59-c1-0-1
Degree $2$
Conductor $117$
Sign $0.0396 - 0.999i$
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  + (−0.530 + 0.530i)2-s + (−1.31 − 1.12i)3-s + 1.43i·4-s + (0.231 + 0.862i)5-s + (1.29 − 0.102i)6-s + (0.858 + 3.20i)7-s + (−1.82 − 1.82i)8-s + (0.472 + 2.96i)9-s + (−0.580 − 0.334i)10-s + (1.22 + 1.22i)11-s + (1.61 − 1.89i)12-s + (0.450 + 3.57i)13-s + (−2.15 − 1.24i)14-s + (0.664 − 1.39i)15-s − 0.937·16-s + (−1.04 − 1.80i)17-s + ⋯
L(s)  = 1  + (−0.375 + 0.375i)2-s + (−0.760 − 0.648i)3-s + 0.718i·4-s + (0.103 + 0.385i)5-s + (0.529 − 0.0419i)6-s + (0.324 + 1.21i)7-s + (−0.644 − 0.644i)8-s + (0.157 + 0.987i)9-s + (−0.183 − 0.105i)10-s + (0.368 + 0.368i)11-s + (0.466 − 0.546i)12-s + (0.124 + 0.992i)13-s + (−0.576 − 0.332i)14-s + (0.171 − 0.360i)15-s − 0.234·16-s + (−0.252 − 0.437i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0396 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0396 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.0396 - 0.999i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.0396 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.494077 + 0.474866i\)
\(L(\frac12)\) \(\approx\) \(0.494077 + 0.474866i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.31 + 1.12i)T \)
13 \( 1 + (-0.450 - 3.57i)T \)
good2 \( 1 + (0.530 - 0.530i)T - 2iT^{2} \)
5 \( 1 + (-0.231 - 0.862i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.858 - 3.20i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.22 - 1.22i)T + 11iT^{2} \)
17 \( 1 + (1.04 + 1.80i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.920 + 3.43i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.52 + 6.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.92iT - 29T^{2} \)
31 \( 1 + (-3.67 + 0.984i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.351 - 1.31i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-7.79 - 2.08i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.50 + 3.75i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.528 + 1.97i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 - 6.71iT - 53T^{2} \)
59 \( 1 + (-1.39 - 1.39i)T + 59iT^{2} \)
61 \( 1 + (0.160 - 0.278i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.07 + 4.01i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.32 + 0.624i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-10.7 + 10.7i)T - 73iT^{2} \)
79 \( 1 + (-2.82 - 4.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.76 - 2.08i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-17.9 + 4.80i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.48 - 0.933i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.68690919408067041454010733249, −12.36816828731994249291245527155, −11.99692963632077277550900858471, −10.90877125769881535320649614988, −9.250547790893227227778374927964, −8.358991618036084087217502572692, −7.01512518739015155486064079283, −6.35927689391475808078247362149, −4.73969387219532699308194926699, −2.46240763826230736449105229799, 1.00908673819291041950505170318, 3.89487067145241129442122944433, 5.27405133345875702549260210058, 6.27974844605595798995729369399, 8.015064084130443841490606384924, 9.474148841975331661185310007347, 10.23724049231714080345825324790, 10.98519041555613316753339590477, 11.85520215843233795783060096706, 13.28465708835525953469722398549

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