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Properties

Label 1-95-95.79-r1-0-0
Degree $1$
Conductor $95$
Sign $0.939 - 0.341i$
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  + (0.173 + 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (0.766 + 0.642i)6-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.766 + 0.642i)13-s + (0.939 + 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + 18-s + (−0.173 − 0.984i)21-s + (0.766 − 0.642i)22-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (0.766 + 0.642i)6-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.766 + 0.642i)13-s + (0.939 + 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + 18-s + (−0.173 − 0.984i)21-s + (0.766 − 0.642i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.939 - 0.341i$
Analytic conductor: \(10.2091\)
Root analytic conductor: \(10.2091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (1:\ ),\ 0.939 - 0.341i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.099023001 - 0.3694081111i\)
\(L(\frac12)\) \(\approx\) \(2.099023001 - 0.3694081111i\)
\(L(1)\) \(\approx\) \(1.421858729 + 0.07750006576i\)
\(L(1)\) \(\approx\) \(1.421858729 + 0.07750006576i\)

Euler product

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

−30.541182797504876964040066966998, −28.70354483705808549040918192103, −28.01565728182748669020159934548, −27.14852620911129502305723306102, −25.97200827657479229387217679715, −24.95158505286936344116618621045, −23.4339731370685970355333315390, −22.291743288070841142520196965430, −21.25293886883521270073841680439, −20.69463610756062607529241886508, −19.579615082551631265369458894182, −18.55208424007533454130299909846, −17.44651211523117486839891275732, −15.46685897399951448647871510857, −14.8953541901138806899684492633, −13.53389267626831317783201045788, −12.53546900750803503576565365498, −11.099150073308508716257034213498, −10.13368708091746674072581395485, −8.94932532310373905004733394107, −8.05265677472669380825734932288, −5.53261961734548365616270152572, −4.3925302826580647141317653687, −3.02004940579589628025304414641, −1.83359593798650760732477226205, 0.911379565306398928369655305141, 3.18027835745301044818227633007, 4.5764281560229992446386055533, 6.28265109558496556166590415692, 7.37012175533149548442046968713, 8.29909850916286147682538216432, 9.36622231922939844268857520005, 11.21973086763165950301861306334, 12.96381541936076682901105833801, 13.75787750010725685825650251115, 14.474380005648325407019843357210, 15.81956902195692515379194373387, 16.90771705222741417844927226795, 18.18703861297390185530206286029, 18.89341701603248496858016829551, 20.427325668844907664164168228932, 21.37142635254126394704349836757, 23.01168598929626473220810600586, 23.84736796551711123793995926542, 24.544878077546222347458524030248, 25.67918837245098955647092305039, 26.53827775388278389424962944709, 27.259845066593016378359594451465, 29.04037156115503409330219332046, 30.19847810436809557943029428003

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