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Properties

Label 2-117-117.50-c1-0-7
Degree $2$
Conductor $117$
Sign $0.607 + 0.794i$
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  + (−0.104 − 0.391i)2-s + (0.735 − 1.56i)3-s + (1.58 − 0.917i)4-s + (0.537 + 2.00i)5-s + (−0.690 − 0.123i)6-s + (−1.39 + 1.39i)7-s + (−1.09 − 1.09i)8-s + (−1.91 − 2.30i)9-s + (0.728 − 0.420i)10-s + (−2.68 + 0.720i)11-s + (−0.270 − 3.16i)12-s + (3.47 − 0.957i)13-s + (0.692 + 0.399i)14-s + (3.54 + 0.632i)15-s + (1.52 − 2.63i)16-s + (−2.88 + 5.00i)17-s + ⋯
L(s)  = 1  + (−0.0741 − 0.276i)2-s + (0.424 − 0.905i)3-s + (0.794 − 0.458i)4-s + (0.240 + 0.897i)5-s + (−0.282 − 0.0503i)6-s + (−0.527 + 0.527i)7-s + (−0.388 − 0.388i)8-s + (−0.639 − 0.768i)9-s + (0.230 − 0.133i)10-s + (−0.810 + 0.217i)11-s + (−0.0781 − 0.914i)12-s + (0.964 − 0.265i)13-s + (0.184 + 0.106i)14-s + (0.914 + 0.163i)15-s + (0.380 − 0.658i)16-s + (−0.700 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.607 + 0.794i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.607 + 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12036 - 0.554003i\)
\(L(\frac12)\) \(\approx\) \(1.12036 - 0.554003i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.735 + 1.56i)T \)
13 \( 1 + (-3.47 + 0.957i)T \)
good2 \( 1 + (0.104 + 0.391i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-0.537 - 2.00i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.39 - 1.39i)T - 7iT^{2} \)
11 \( 1 + (2.68 - 0.720i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.88 - 5.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.00 - 1.07i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 2.93T + 23T^{2} \)
29 \( 1 + (0.0236 + 0.0136i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.49 - 1.20i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-6.63 - 1.77i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-6.64 + 6.64i)T - 41iT^{2} \)
43 \( 1 - 6.27iT - 43T^{2} \)
47 \( 1 + (1.14 - 4.26i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + 8.65iT - 53T^{2} \)
59 \( 1 + (-3.56 + 13.2i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 9.28T + 61T^{2} \)
67 \( 1 + (4.68 + 4.68i)T + 67iT^{2} \)
71 \( 1 + (2.46 + 9.20i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.696 + 0.696i)T - 73iT^{2} \)
79 \( 1 + (6.46 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.21 - 0.860i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-0.569 + 2.12i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-8.64 - 8.64i)T + 97iT^{2} \)
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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

−13.11384563351456860009877246994, −12.57652545324340862915523905621, −11.11862459998043485600114418798, −10.54370112925204189045102251989, −9.128336467235363134929496126871, −7.82795984111081795469267199252, −6.50658094405503234418677971907, −6.05673780334691000406803184888, −3.16328374863487413581451707055, −2.07567010018075929808332385626, 2.76341638020609746313259557819, 4.28806953113095585957754948384, 5.72615184909550819072680194528, 7.20048984497195776981508238837, 8.498690935312156502569185042927, 9.225360770364121439488280383795, 10.62125915061154681465622667475, 11.36328590708868133970455129311, 12.93761273464113870854665509660, 13.52796174747060265484302805740

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