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Properties

Label 1-6069-6069.188-r0-0-0
Degree $1$
Conductor $6069$
Sign $0.906 + 0.421i$
Analytic cond. $28.1843$
Root an. cond. $28.1843$
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  + (0.273 − 0.961i)2-s + (−0.850 − 0.526i)4-s + (0.739 + 0.673i)5-s + (−0.739 + 0.673i)8-s + (0.850 − 0.526i)10-s + (0.850 + 0.526i)11-s + (−0.739 + 0.673i)13-s + (0.445 + 0.895i)16-s + (0.273 + 0.961i)19-s + (−0.273 − 0.961i)20-s + (0.739 − 0.673i)22-s + (0.982 + 0.183i)23-s + (0.0922 + 0.995i)25-s + (0.445 + 0.895i)26-s + (0.850 + 0.526i)29-s + ⋯
L(s)  = 1  + (0.273 − 0.961i)2-s + (−0.850 − 0.526i)4-s + (0.739 + 0.673i)5-s + (−0.739 + 0.673i)8-s + (0.850 − 0.526i)10-s + (0.850 + 0.526i)11-s + (−0.739 + 0.673i)13-s + (0.445 + 0.895i)16-s + (0.273 + 0.961i)19-s + (−0.273 − 0.961i)20-s + (0.739 − 0.673i)22-s + (0.982 + 0.183i)23-s + (0.0922 + 0.995i)25-s + (0.445 + 0.895i)26-s + (0.850 + 0.526i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6069\)    =    \(3 \cdot 7 \cdot 17^{2}\)
Sign: $0.906 + 0.421i$
Analytic conductor: \(28.1843\)
Root analytic conductor: \(28.1843\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6069} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6069,\ (0:\ ),\ 0.906 + 0.421i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.053489286 + 0.4536215178i\)
\(L(\frac12)\) \(\approx\) \(2.053489286 + 0.4536215178i\)
\(L(1)\) \(\approx\) \(1.264809567 - 0.2475341872i\)
\(L(1)\) \(\approx\) \(1.264809567 - 0.2475341872i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.273 - 0.961i)T \)
5 \( 1 + (0.739 + 0.673i)T \)
11 \( 1 + (0.850 + 0.526i)T \)
13 \( 1 + (-0.739 + 0.673i)T \)
19 \( 1 + (0.273 + 0.961i)T \)
23 \( 1 + (0.982 + 0.183i)T \)
29 \( 1 + (0.850 + 0.526i)T \)
31 \( 1 + (-0.739 + 0.673i)T \)
37 \( 1 + (-0.602 - 0.798i)T \)
41 \( 1 + (0.0922 - 0.995i)T \)
43 \( 1 + (0.445 - 0.895i)T \)
47 \( 1 + (-0.982 + 0.183i)T \)
53 \( 1 + (0.982 + 0.183i)T \)
59 \( 1 + (0.932 - 0.361i)T \)
61 \( 1 + (-0.932 - 0.361i)T \)
67 \( 1 + (-0.273 - 0.961i)T \)
71 \( 1 + (0.982 + 0.183i)T \)
73 \( 1 + (-0.445 - 0.895i)T \)
79 \( 1 + (-0.273 - 0.961i)T \)
83 \( 1 + (0.0922 + 0.995i)T \)
89 \( 1 + (0.739 + 0.673i)T \)
97 \( 1 + (0.982 + 0.183i)T \)
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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

−17.46019580735158556297071118976, −16.97641571397252552395207464516, −16.42417666824303494423178584828, −15.76763848714676380878759277646, −14.8801031023488703222294886305, −14.56953031715300987746570290555, −13.61567373725324633371976088946, −13.28440395239589871802634598912, −12.63864184302262879304922279950, −11.90763470006232077486007034159, −11.16247979804945648986786714661, −9.95867646616152167746381248436, −9.62984577618575899441800181949, −8.757297631971056472318740962158, −8.44653908825920755237731339084, −7.470239803952861837773451160023, −6.799601664640900374319059257460, −6.1283579321445711302266983046, −5.47969710252063042594717573884, −4.82201085990035211180588470865, −4.29845342101891893886696743504, −3.200381945297550741724556625768, −2.5890192452485094168460756954, −1.27563771099391919266595318230, −0.5318500335032533456441882225, 1.0673046206734815423275105350, 1.879182464431039819141004567206, 2.29590873357909806494956614520, 3.35864538338188289356922491017, 3.765841282144580726559224706394, 4.83754147490732480122166505943, 5.30751339091831827298047648382, 6.22260185091995914416503966054, 6.881669932481464757896483405462, 7.563039699379525032029440238499, 8.90197375223650336706483224305, 9.15724931075949910218206992782, 9.925336687683982074273356026200, 10.49527791109906459295214498004, 11.04738562703081815063082212091, 12.01597277203098022132679974519, 12.25283298190563251202335716217, 13.10820979983015283403496582637, 13.897115079314872181224597457180, 14.44275801902423223855125650644, 14.649281170019463384872500337862, 15.604350067276568897657260012168, 16.692018951522029810123358539231, 17.2639517139087294535628631735, 17.85548019021814609979168069910

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