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Properties

Label 2-117-117.11-c1-0-2
Degree $2$
Conductor $117$
Sign $0.450 - 0.892i$
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  + (0.375 − 0.375i)2-s + (−0.0440 + 1.73i)3-s + 1.71i·4-s + (−2.85 − 0.764i)5-s + (0.633 + 0.667i)6-s + (3.85 + 1.03i)7-s + (1.39 + 1.39i)8-s + (−2.99 − 0.152i)9-s + (−1.35 + 0.784i)10-s + (1.41 + 1.41i)11-s + (−2.97 − 0.0757i)12-s + (−0.867 − 3.49i)13-s + (1.83 − 1.05i)14-s + (1.44 − 4.90i)15-s − 2.38·16-s + (2.38 − 4.12i)17-s + ⋯
L(s)  = 1  + (0.265 − 0.265i)2-s + (−0.0254 + 0.999i)3-s + 0.858i·4-s + (−1.27 − 0.341i)5-s + (0.258 + 0.272i)6-s + (1.45 + 0.390i)7-s + (0.493 + 0.493i)8-s + (−0.998 − 0.0508i)9-s + (−0.429 + 0.248i)10-s + (0.428 + 0.428i)11-s + (−0.858 − 0.0218i)12-s + (−0.240 − 0.970i)13-s + (0.490 − 0.283i)14-s + (0.374 − 1.26i)15-s − 0.596·16-s + (0.577 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.450 - 0.892i$
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: \((2,\ 117,\ (\ :1/2),\ 0.450 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.939932 + 0.578534i\)
\(L(\frac12)\) \(\approx\) \(0.939932 + 0.578534i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0440 - 1.73i)T \)
13 \( 1 + (0.867 + 3.49i)T \)
good2 \( 1 + (-0.375 + 0.375i)T - 2iT^{2} \)
5 \( 1 + (2.85 + 0.764i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-3.85 - 1.03i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.41 - 1.41i)T + 11iT^{2} \)
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 \( 1 + (0.469 - 0.812i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.41iT - 29T^{2} \)
31 \( 1 + (-0.0175 + 0.0653i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.00215 + 0.000576i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.08 - 7.77i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (8.64 - 4.99i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.69 + 1.52i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 7.99iT - 53T^{2} \)
59 \( 1 + (3.08 + 3.08i)T + 59iT^{2} \)
61 \( 1 + (6.15 + 10.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.05 + 0.282i)T + (58.0 - 33.5i)T^{2} \)
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 \( 1 + (-2.06 - 7.70i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-0.189 + 0.708i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.36 + 5.11i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.81533768439071521526580892970, −12.17335063934245424915782162336, −11.74642603103827370190573369958, −11.01078176965640963132008041378, −9.408279017897609053891446867420, −8.119230309631761807304332003585, −7.72781150062762627889927311392, −5.10733100054584930218062129094, −4.40140465889313103767997190962, −3.10177504672045394850138986062, 1.45878292505398273536679279012, 4.01352194994265877533215451198, 5.44057095585180127814355752953, 6.87700110979854852072995322693, 7.65492901892683715858986221782, 8.697511647547752058966101291362, 10.60581751924078597260850553884, 11.45891631460153480397516339162, 12.09121508498959308312894972671, 13.69632202817163985954759044000

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