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Properties

Label 1-315-315.23-r0-0-0
Degree $1$
Conductor $315$
Sign $-0.206 + 0.978i$
Analytic cond. $1.46285$
Root an. cond. $1.46285$
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 − 4-s − i·8-s + (0.5 + 0.866i)11-s + (0.866 − 0.5i)13-s + 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 + (0.5 + 0.866i)26-s + (−0.5 + 0.866i)29-s + 31-s + i·32-s + (0.5 − 0.866i)34-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·8-s + (0.5 + 0.866i)11-s + (0.866 − 0.5i)13-s + 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 + (0.5 + 0.866i)26-s + (−0.5 + 0.866i)29-s + 31-s + i·32-s + (0.5 − 0.866i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.206 + 0.978i$
Analytic conductor: \(1.46285\)
Root analytic conductor: \(1.46285\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 315,\ (0:\ ),\ -0.206 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7217936362 + 0.8899689554i\)
\(L(\frac12)\) \(\approx\) \(0.7217936362 + 0.8899689554i\)
\(L(1)\) \(\approx\) \(0.8530322037 + 0.5443723488i\)
\(L(1)\) \(\approx\) \(0.8530322037 + 0.5443723488i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + iT \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
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 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 - iT \)
71 \( 1 - T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 - T \)
83 \( 1 + (0.866 + 0.5i)T \)
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

−24.796995411636336890101578308793, −23.94710866835040137627163098024, −22.9326252089589999733267458500, −22.12788562316986278409334941679, −21.30554061657037472245913826055, −20.55648715127750615378583380095, −19.453333571421430338874396818764, −18.95201688067416005459786244785, −17.86668939876561850351090064088, −17.050571755455825643787948621975, −15.89307353379866131517852900301, −14.68196476975508267801704290195, −13.6079925916273293534655994145, −13.122291273080249015943462360812, −11.70851234692257293939891289800, −11.22579852840249796574327355922, −10.20682949194561450726622095652, −8.94353074649545795241187401777, −8.52172250404261427887825829744, −6.84487680986409468277258075917, −5.6465637253357673429978719564, −4.376879965414002127981865011706, −3.47010270691730208098391282517, −2.2458774081761730749034847594, −0.89405535088899328631876332263, 1.33416097625000478035901840005, 3.288582721407186152465211872879, 4.4371156588480225277900067040, 5.44854527383416977517325769069, 6.55391673231842232761218344442, 7.37735262554913232151876297054, 8.48111497779564262462095378107, 9.35379149415952482161011420445, 10.35592559098165009358819821759, 11.713447471948172617061541942995, 12.87306800282808221084413958915, 13.65910830292520782902764036202, 14.68436967278062272602552803322, 15.47629612203463826781232860290, 16.287075851299769722179848979241, 17.33970499930383860161297241261, 18.00673899990994543577989409043, 18.900135645207868104260662399593, 20.07821126407811417089665697627, 21.023880525821934946487532497518, 22.33145288287641205682846646977, 22.81533145046609117629586205718, 23.68017085568823326123067997141, 24.80462588617532304287290311491, 25.25885691287386524789949023132

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