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Properties

Label 1-115-115.22-r0-0-0
Degree $1$
Conductor $115$
Sign $0.525 - 0.850i$
Analytic cond. $0.534057$
Root an. cond. $0.534057$
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  + i·2-s − i·3-s − 4-s + 6-s − i·7-s − i·8-s − 9-s − 11-s + i·12-s − i·13-s + 14-s + 16-s − i·17-s − i·18-s + 19-s + ⋯
L(s)  = 1  + i·2-s − i·3-s − 4-s + 6-s − i·7-s − i·8-s − 9-s − 11-s + i·12-s − i·13-s + 14-s + 16-s − i·17-s − i·18-s + 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(0.534057\)
Root analytic conductor: \(0.534057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (0:\ ),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6680381980 - 0.3724556893i\)
\(L(\frac12)\) \(\approx\) \(0.6680381980 - 0.3724556893i\)
\(L(1)\) \(\approx\) \(0.8376714048 - 0.09225174645i\)
\(L(1)\) \(\approx\) \(0.8376714048 - 0.09225174645i\)

Euler product

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

−28.99819858533589332646767839057, −28.515484614809218202289071197159, −27.70174485919453748489302338730, −26.49685653676437497729135502620, −25.93388512340070286242862786454, −24.16199318686836411486966290520, −22.9271873010416688595518729477, −21.91406225194034184465599426199, −21.30910490505895811992861040487, −20.45226646819176854516971327212, −19.201694731993165798260241153277, −18.29420325904321819873778283244, −17.03138580631204901313173350205, −15.74314974170116751597007975572, −14.72684208729019894125316226329, −13.51042252099451692295443558719, −12.14728208423837505923693864415, −11.2519960479641007099008479372, −10.13240251510197558162710585885, −9.200781289430932165609533912083, −8.214447764703188014167123671119, −5.75934672464623345776334564983, −4.756924298910491085952422347461, −3.4228681201039012030788898729, −2.199928185192084790206002843524, 0.75662131039019637418656834919, 3.09696311304298618561943045470, 4.94469822878918910700773224126, 6.08719723881811895872606309348, 7.562523350734194193217000182364, 7.72790579390347925440623237680, 9.432156360914694131983218481022, 10.85441624329397541338059841897, 12.5524703223377849333060986834, 13.46069389830555538903859332770, 14.16008527753850860321689838451, 15.57295762084771910378951577165, 16.66305630292418669214073005375, 17.782827719357717093912055452383, 18.3329463396254463905274650213, 19.65635769586867918714568215770, 20.76139095361766182579602955350, 22.68320983016541875822393794004, 23.06934129413658828145501367714, 24.19830976724080510721335422454, 24.85939646597676013289397506239, 26.01188823825516679106428362211, 26.72386758549528447153031043612, 28.01988318105173831797997385330, 29.2459253737005101838727436373

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