Properties

Label 1-195-195.23-r0-0-0
Degree $1$
Conductor $195$
Sign $0.536 + 0.843i$
Analytic cond. $0.905576$
Root an. cond. $0.905576$
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)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)7-s + i·8-s + (−0.5 + 0.866i)11-s + 14-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + (−0.866 + 0.5i)22-s + (0.866 + 0.5i)23-s + (0.866 + 0.5i)28-s + (−0.5 + 0.866i)29-s − 31-s + (−0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)7-s + i·8-s + (−0.5 + 0.866i)11-s + 14-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + (−0.866 + 0.5i)22-s + (0.866 + 0.5i)23-s + (0.866 + 0.5i)28-s + (−0.5 + 0.866i)29-s − 31-s + (−0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.536 + 0.843i$
Analytic conductor: \(0.905576\)
Root analytic conductor: \(0.905576\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 195,\ (0:\ ),\ 0.536 + 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.735525730 + 0.9530837438i\)
\(L(\frac12)\) \(\approx\) \(1.735525730 + 0.9530837438i\)
\(L(1)\) \(\approx\) \(1.620050086 + 0.5882352814i\)
\(L(1)\) \(\approx\) \(1.620050086 + 0.5882352814i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 - iT \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 + iT \)
89 \( 1 + (0.5 - 0.866i)T \)
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

−27.15042030010197476961649012655, −25.68662025927038289628853346799, −24.64171421579552876523877646039, −23.95470268263317914466146528857, −23.05240132896572037477873306781, −21.96510228032533054741410346477, −21.08931239304821977636081278600, −20.63360663698645416924492085189, −19.042546686320868596482942354761, −18.685884172808354275244286999698, −17.13105866964184176029238920526, −15.95106619036529320961619702431, −14.87265946713106394492543813952, −14.21754616408542764065186109493, −13.03781578248390279464843771527, −12.12529376959329520695623600925, −11.11629542976639580408978319856, −10.324149037223248737624161492618, −8.82813859241131379367899309235, −7.62865328891577285366623647135, −6.02167074678789012462387609508, −5.30075460252508615551164032449, −4.010712955643669202729737974467, −2.76249746267735420758618157493, −1.469804954360373472381151485120, 1.918755483050532548837690289228, 3.382822225506633739312129667852, 4.699594662246284629569045990574, 5.38268237194734773501175386890, 7.035837712678214350069147160062, 7.59387301475659339787111976909, 8.91662594064037101680902314062, 10.53990883848777474286520808700, 11.50460433917584366489389245734, 12.61428367460888588426133325873, 13.52724468800840131938709025752, 14.58485730445865791390385587933, 15.23856537949536327635447108992, 16.44954144300579526881042076374, 17.350556636977567055265043522332, 18.21863551675562495381459866131, 19.84612990029558441990130664439, 20.76461178745947382116878015020, 21.405269748069399994631275025278, 22.615716574521265493855748807731, 23.48538697126580967172164620104, 24.05982937676700176674865587808, 25.24830535005395390282864745358, 25.9187409216765427597603920433, 27.02412027370164647525635252885

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