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Properties

Label 1-115-115.68-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: $\chi_{115} (68, \cdot )$
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

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

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