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Properties

Label 1-315-315.83-r1-0-0
Degree $1$
Conductor $315$
Sign $0.313 - 0.949i$
Analytic cond. $33.8514$
Root an. cond. $33.8514$
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 − i·8-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + (−0.5 − 0.866i)16-s − i·17-s + 19-s + (0.866 + 0.5i)22-s + (0.866 + 0.5i)23-s + 26-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.866 − 0.5i)32-s + (−0.5 − 0.866i)34-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s − i·8-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + (−0.5 − 0.866i)16-s − i·17-s + 19-s + (0.866 + 0.5i)22-s + (0.866 + 0.5i)23-s + 26-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.866 − 0.5i)32-s + (−0.5 − 0.866i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.313 - 0.949i$
Analytic conductor: \(33.8514\)
Root analytic conductor: \(33.8514\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 315,\ (1:\ ),\ 0.313 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.010532673 - 2.175959747i\)
\(L(\frac12)\) \(\approx\) \(3.010532673 - 2.175959747i\)
\(L(1)\) \(\approx\) \(1.822180171 - 0.7394867763i\)
\(L(1)\) \(\approx\) \(1.822180171 - 0.7394867763i\)

Euler product

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

−24.955303025218697459820116667913, −24.253508753228150530349825956609, −23.392757829160327759715952498516, −22.508492867654009647429801938089, −21.76615790251536824342224600932, −20.873101343123735700507240235371, −20.020643192996052414128067195216, −18.86174889984880645333085592980, −17.72044409670545464334818332722, −16.774083802864518914364081979898, −15.99814155142319662112683742541, −15.08772997210167775435266435639, −14.15032315738423441058713092367, −13.35381092090054208941635744372, −12.45950202036333767750164943473, −11.402465112969827503998767824629, −10.576037038656130406159054763795, −8.888965959984905068414695564563, −8.15166451759268803295500067591, −6.90324663056704399611576487659, −6.01667485025386578555294638577, −5.09786779236437078245958046060, −3.76278982149753374111925319162, −3.02556394933161243033801243378, −1.27977042877722272810965302501, 0.95260812560668794687253241836, 2.16003494704414309310245083407, 3.42167276518588409328023193339, 4.41983650222700807084497467469, 5.442539463394969527365350642711, 6.57602689362014382674084724680, 7.49805711729105850541207162437, 9.22406389395186774012075840125, 9.88627116515102953405858621922, 11.30522987271227213361896769999, 11.71412175301484950223560243662, 12.932794026916953007574636577167, 13.70812181016419532672010598135, 14.593493066639245631796846135751, 15.54455366659505037832163248412, 16.36950737396642338183699063671, 17.70671483943135466922368243654, 18.7091307895349342432774106163, 19.58289420227588962462636353162, 20.62231190513259221813834510400, 21.02642771199050660990463862052, 22.36118158304336399082461509141, 22.80151688161929778302167150820, 23.71527845789831079914148672097, 24.7318039932553571225843910355

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