↑

Properties

Label 1-95-95.69-r1-0-0
Degree $1$
Conductor $95$
Sign $0.910 + 0.412i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)6-s − 7-s + 8-s + (−0.5 − 0.866i)9-s + 11-s + 12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + 18-s + (0.5 − 0.866i)21-s + (−0.5 + 0.866i)22-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)6-s − 7-s + 8-s + (−0.5 − 0.866i)9-s + 11-s + 12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + 18-s + (0.5 − 0.866i)21-s + (−0.5 + 0.866i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.910 + 0.412i$
Analytic conductor: \(10.2091\)
Root analytic conductor: \(10.2091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (1:\ ),\ 0.910 + 0.412i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7493325977 + 0.1619510970i\)
\(L(\frac12)\) \(\approx\) \(0.7493325977 + 0.1619510970i\)
\(L(1)\) \(\approx\) \(0.5871051878 + 0.2662138030i\)
\(L(1)\) \(\approx\) \(0.5871051878 + 0.2662138030i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.74966261505731444281302099449, −28.83894445812640638535875266831, −28.20626795668210430280522992642, −26.89997357630257977515182690369, −25.76949348832090165346543984798, −24.78590510717837387987223039755, −23.33496306201211115476656486483, −22.41049614845421594339612838109, −21.53831678171901603493006431512, −19.83373899034294888240598593538, −19.286058874785705998691599616115, −18.3887552136322795391384216772, −17.026946294358975562711379925390, −16.55723773771032536640535227131, −14.32218177708608255235017633188, −13.03576887056176117008546252422, −12.27556202285972215033199254752, −11.29525380683775317781212120902, −9.957405876840713118717771544893, −8.79022316154758060202808409492, −7.32504309729048522299432997256, −6.20171853055659209316800961221, −4.18930032711182666460589212286, −2.550530449512920603019314588028, −1.060958596821136263492070983440, 0.55805807764059027320931887862, 3.48658840108925729744621219309, 5.05189825357740803548480255399, 6.11573377234112734838858333664, 7.30923577058409387829414276624, 9.118433287416195014042408156910, 9.711674132828103668734264782389, 10.91055972367194092521789613937, 12.47033963688025007383484740907, 14.11580275800595363554161141214, 15.196946149288540004136961562881, 16.13494534404999995668951490903, 16.92688130404687464087442286197, 17.90256257262080779985128565353, 19.31351249207235006511626653061, 20.27872269258700822284501091280, 22.0515037880881388322529285260, 22.640610059556049767791609854250, 23.63422119695295554647229646289, 25.13997025493791388024563756041, 25.758261657386634641099952199355, 27.22069767797567502113412896233, 27.42091028249805495963153496881, 28.778452762143196522714278020939, 29.58405930363759882008322444684

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