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Properties

Label 1-468-468.115-r0-0-0
Degree $1$
Conductor $468$
Sign $0.612 + 0.790i$
Analytic cond. $2.17338$
Root an. cond. $2.17338$
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  + (−0.866 − 0.5i)5-s + (−0.866 − 0.5i)7-s − i·11-s + (0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.5 − 0.866i)23-s + (0.5 + 0.866i)25-s + 29-s + (0.866 + 0.5i)31-s + (0.5 + 0.866i)35-s + (0.866 + 0.5i)37-s + (0.866 − 0.5i)41-s + (−0.5 + 0.866i)43-s + (0.866 − 0.5i)47-s + (0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)5-s + (−0.866 − 0.5i)7-s − i·11-s + (0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.5 − 0.866i)23-s + (0.5 + 0.866i)25-s + 29-s + (0.866 + 0.5i)31-s + (0.5 + 0.866i)35-s + (0.866 + 0.5i)37-s + (0.866 − 0.5i)41-s + (−0.5 + 0.866i)43-s + (0.866 − 0.5i)47-s + (0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.612 + 0.790i$
Analytic conductor: \(2.17338\)
Root analytic conductor: \(2.17338\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 468,\ (0:\ ),\ 0.612 + 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7307205106 + 0.3580743777i\)
\(L(\frac12)\) \(\approx\) \(0.7307205106 + 0.3580743777i\)
\(L(1)\) \(\approx\) \(0.8031793198 + 0.04865367822i\)
\(L(1)\) \(\approx\) \(0.8031793198 + 0.04865367822i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 - iT \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + T \)
59 \( 1 - iT \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−23.53517199163784442665498678749, −23.01507813122017555555451961755, −22.00865670921567964426595014284, −21.44375441670244655084479710437, −20.13480038968324204315908711651, −19.27950190759781637733459761396, −18.88874326486178787150069913454, −17.9065232828377677364200937458, −16.63597094209213301948311053329, −15.89997644502956020504226141192, −15.30082949652061098775677698258, −14.19304005607385235897147386655, −13.3197187254039970659731468250, −12.20903038219655288228952794268, −11.53654025411286604296074464579, −10.59563352696078726468780600233, −9.53680382513130127766887876438, −8.55771678279225590123539998757, −7.62561824960623199801477797843, −6.57631448170124113769105001864, −5.76915104920007460159822353954, −4.37503082988750831705949101589, −3.29629380794678055153346939990, −2.58397393837683648934193656431, −0.53542920986895858854598697817, 1.119978928463557079279523896076, 2.69157102159371486247757858726, 3.993926581185943803481580411423, 4.49795220725772240320313337276, 6.00701590902740384950338899814, 6.953660277522845195073702882884, 7.923352661064064021068673655711, 8.76345330829754931358869562867, 10.00253869806488884882824288690, 10.57939729812393049532495559728, 12.06214514011366928949755068116, 12.47053438697529089088476923709, 13.373186567865993937468831595118, 14.633075517150705812374674675205, 15.36720106821073198730628711630, 16.3567921849948050940028775592, 16.88484012154001881749049663372, 17.97892675850707659135832544412, 19.17677701391436988946326669532, 19.654119605754151735238330710626, 20.46037908197803716476510152989, 21.33547865812743825825361827505, 22.5737290124973908509274380569, 23.19116242627947143218454596811, 23.744882795232745322648341036197

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