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Properties

Label 1-95-95.82-r1-0-0
Degree $1$
Conductor $95$
Sign $0.873 - 0.487i$
Analytic cond. $10.2091$
Root an. cond. $10.2091$
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  + (0.984 + 0.173i)2-s + (0.642 − 0.766i)3-s + (0.939 + 0.342i)4-s + (0.766 − 0.642i)6-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s + (−0.642 − 0.766i)13-s + (0.939 − 0.342i)14-s + (0.766 + 0.642i)16-s + (0.984 + 0.173i)17-s − i·18-s + (0.173 − 0.984i)21-s + (−0.642 + 0.766i)22-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (0.642 − 0.766i)3-s + (0.939 + 0.342i)4-s + (0.766 − 0.642i)6-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s + (−0.642 − 0.766i)13-s + (0.939 − 0.342i)14-s + (0.766 + 0.642i)16-s + (0.984 + 0.173i)17-s − i·18-s + (0.173 − 0.984i)21-s + (−0.642 + 0.766i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.873 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.873 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.873 - 0.487i$
Analytic conductor: \(10.2091\)
Root analytic conductor: \(10.2091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (1:\ ),\ 0.873 - 0.487i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.879153970 - 1.008662852i\)
\(L(\frac12)\) \(\approx\) \(3.879153970 - 1.008662852i\)
\(L(1)\) \(\approx\) \(2.414037610 - 0.3786278221i\)
\(L(1)\) \(\approx\) \(2.414037610 - 0.3786278221i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.984 + 0.173i)T \)
3 \( 1 + (0.642 - 0.766i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.642 - 0.766i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (-0.984 + 0.173i)T \)
53 \( 1 + (-0.342 + 0.939i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.642 - 0.766i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.984 + 0.173i)T \)
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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

−30.35507232720587853240979978556, −29.11595051824053979153169099696, −27.98093875898855011001213465691, −26.908080328577975043679499447613, −25.721047102624776256215287912278, −24.627017220605518403699239514428, −23.83265549903880146998250446090, −22.35395582311919393287923980186, −21.40415168921274021018278397082, −20.95934562098701735885440110443, −19.71038851118914437629292092011, −18.62847392854079064125131107281, −16.631101893604957229739292396538, −15.80680587362641595827843629057, −14.42936380510185449462220462985, −14.2251706781773141033485918794, −12.57792152240244386504874461540, −11.32626887708824569369743513246, −10.33829763851982138443980818420, −8.808534235822728137106897750036, −7.500613852253286327677286821340, −5.59632450785665200352903515368, −4.67658831945540031462427970070, −3.29820349004414119136071395334, −2.05152025849337246901375094102, 1.594104466949408611198744010629, 2.94042072090956638991781672363, 4.45246959165343248867336126382, 5.85445492495858179680753700602, 7.563926930717665818275263300667, 7.79567702098012848536681443950, 9.977889030074271093572859079603, 11.55132064617326635038935783713, 12.60214676733683613205251927819, 13.52855083962099343562903492937, 14.617264325845588608774835279643, 15.26409712269098395902227332199, 17.02618246940486342704473168927, 18.00297421793338313781442131760, 19.573727517743702451464162094200, 20.46489371589819485328127718660, 21.22104826855390283312097516641, 22.80617883979488516357759142476, 23.643082208225774233695692793348, 24.48777008375268519433589421590, 25.42361963274336171469293428154, 26.32142437792865592805740261683, 27.84690100041170690141763832017, 29.42593134479340744786003827428, 30.04759621799614061311053460642

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