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Properties

Label 1-119-119.118-r1-0-0
Degree $1$
Conductor $119$
Sign $1$
Analytic cond. $12.7883$
Root an. cond. $12.7883$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 11-s + 12-s − 13-s + 15-s + 16-s + 18-s − 19-s + 20-s − 22-s − 23-s + 24-s + 25-s − 26-s + 27-s − 29-s + 30-s + 31-s + 32-s − 33-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 11-s + 12-s − 13-s + 15-s + 16-s + 18-s − 19-s + 20-s − 22-s − 23-s + 24-s + 25-s − 26-s + 27-s − 29-s + 30-s + 31-s + 32-s − 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $1$
Analytic conductor: \(12.7883\)
Root analytic conductor: \(12.7883\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{119} (118, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.011534525\)
\(L(\frac12)\) \(\approx\) \(5.011534525\)
\(L(1)\) \(\approx\) \(2.879893263\)
\(L(1)\) \(\approx\) \(2.879893263\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
17 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + T \)
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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

−29.41838352243084177795807653505, −28.0572732934585955344870530796, −26.28078983079156336165324605519, −25.82046918559971302493406943393, −24.66306166251892801915599076345, −24.09834464833981789047805299890, −22.61565571824615385561458050011, −21.4826232502967046342719948899, −20.99566036127689175932664659127, −19.938138528775194240790732876097, −18.83417004275596420162824488760, −17.39037758631226144108487633870, −16.035365623376936922626212495764, −14.94745027609096690126945899059, −14.13128831071576937255351546086, −13.218411984485733997958699369956, −12.44005886620115575913559694795, −10.60688236262636842003387564316, −9.720871089420957612112684653400, −8.12704855486758789589290683765, −6.93722723484947764710635847749, −5.5695297606663213515331699030, −4.331965287141966505417937103150, −2.7548590764128740346472305262, −1.98378068595255327055129969058, 1.98378068595255327055129969058, 2.7548590764128740346472305262, 4.331965287141966505417937103150, 5.5695297606663213515331699030, 6.93722723484947764710635847749, 8.12704855486758789589290683765, 9.720871089420957612112684653400, 10.60688236262636842003387564316, 12.44005886620115575913559694795, 13.218411984485733997958699369956, 14.13128831071576937255351546086, 14.94745027609096690126945899059, 16.035365623376936922626212495764, 17.39037758631226144108487633870, 18.83417004275596420162824488760, 19.938138528775194240790732876097, 20.99566036127689175932664659127, 21.4826232502967046342719948899, 22.61565571824615385561458050011, 24.09834464833981789047805299890, 24.66306166251892801915599076345, 25.82046918559971302493406943393, 26.28078983079156336165324605519, 28.0572732934585955344870530796, 29.41838352243084177795807653505

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