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Properties

Label 1-116-116.19-r0-0-0
Degree $1$
Conductor $116$
Sign $0.549 + 0.835i$
Analytic cond. $0.538701$
Root an. cond. $0.538701$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.781 + 0.623i)3-s + (0.900 + 0.433i)5-s + (−0.623 + 0.781i)7-s + (0.222 + 0.974i)9-s + (−0.974 − 0.222i)11-s + (0.222 − 0.974i)13-s + (0.433 + 0.900i)15-s − i·17-s + (−0.781 + 0.623i)19-s + (−0.974 + 0.222i)21-s + (0.900 − 0.433i)23-s + (0.623 + 0.781i)25-s + (−0.433 + 0.900i)27-s + (0.433 − 0.900i)31-s + (−0.623 − 0.781i)33-s + ⋯
L(s)  = 1  + (0.781 + 0.623i)3-s + (0.900 + 0.433i)5-s + (−0.623 + 0.781i)7-s + (0.222 + 0.974i)9-s + (−0.974 − 0.222i)11-s + (0.222 − 0.974i)13-s + (0.433 + 0.900i)15-s − i·17-s + (−0.781 + 0.623i)19-s + (−0.974 + 0.222i)21-s + (0.900 − 0.433i)23-s + (0.623 + 0.781i)25-s + (−0.433 + 0.900i)27-s + (0.433 − 0.900i)31-s + (−0.623 − 0.781i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $0.549 + 0.835i$
Analytic conductor: \(0.538701\)
Root analytic conductor: \(0.538701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 116,\ (0:\ ),\ 0.549 + 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.206818271 + 0.6509206260i\)
\(L(\frac12)\) \(\approx\) \(1.206818271 + 0.6509206260i\)
\(L(1)\) \(\approx\) \(1.264545897 + 0.4106362027i\)
\(L(1)\) \(\approx\) \(1.264545897 + 0.4106362027i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.781 + 0.623i)T \)
5 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (-0.623 + 0.781i)T \)
11 \( 1 + (-0.974 - 0.222i)T \)
13 \( 1 + (0.222 - 0.974i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.781 + 0.623i)T \)
23 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (0.433 - 0.900i)T \)
37 \( 1 + (0.974 - 0.222i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.433 - 0.900i)T \)
47 \( 1 + (0.974 + 0.222i)T \)
53 \( 1 + (-0.900 - 0.433i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.781 - 0.623i)T \)
67 \( 1 + (-0.222 - 0.974i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.433 - 0.900i)T \)
79 \( 1 + (0.974 - 0.222i)T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (-0.433 + 0.900i)T \)
97 \( 1 + (-0.781 + 0.623i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.12232853403617292570308546158, −28.45920805875764690084520571503, −26.62588239115372043347162838441, −25.9305260597220464633000794964, −25.25722225324195867578492232308, −23.917739474494237375060925772784, −23.38941255204886916200934475658, −21.58832024212547485806777625634, −20.86446559294860737461836676169, −19.77707687714650331592659531523, −18.88610268611430519651993070841, −17.677709503088905196696103545412, −16.74589325284494963024865151421, −15.3560661783071072635735492918, −14.030085389886071985463534591063, −13.251430371889636227436981625816, −12.59168627225361184444625365537, −10.6744218701171439200991943559, −9.53475485346699337549651724023, −8.55097368255694176618771239167, −7.16239332530624495474703872813, −6.167245757306882860466035362101, −4.39734124220166609370040582576, −2.81213212148423304380417452767, −1.47143539091173174534532562852, 2.42242367969501546389788081336, 3.1224173411343807647419842721, 5.0376758913337179254423653915, 6.13648167867790519510233980141, 7.818731814128857771466764067856, 9.04225449557193498935705104263, 9.975088167380597151323198533820, 10.870383535573486184734724669383, 12.807055493093048264756286569830, 13.57467290247155679422352468356, 14.843082957491230287761539230955, 15.60326603090487840730206520966, 16.75273423494897768367429223628, 18.30591274925248050206625203552, 18.92802029506900136100156163193, 20.41180556438346381994384798194, 21.17636030363839202402778934939, 22.091766131639807489626284932573, 22.999518424128082208883081549088, 24.9181003286564436119642564086, 25.314227675260146103123654107685, 26.2516974214884696206504370510, 27.213831555662457760761022531408, 28.42577762825719052537990364204, 29.37081166696749394627450672000

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