↑

Properties

Label 1-315-315.202-r0-0-0
Degree $1$
Conductor $315$
Sign $0.949 - 0.313i$
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

Related objects

Downloads

Learn more

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

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

Invariants

Degree: \(1\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.949 - 0.313i$
Analytic conductor: \(1.46285\)
Root analytic conductor: \(1.46285\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 315,\ (0:\ ),\ 0.949 - 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8386146233 - 0.1349438122i\)
\(L(\frac12)\) \(\approx\) \(0.8386146233 - 0.1349438122i\)
\(L(1)\) \(\approx\) \(0.7575030880 - 0.1146677190i\)
\(L(1)\) \(\approx\) \(0.7575030880 - 0.1146677190i\)

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 \( 1 + (0.866 + 0.5i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.43199919734810457358752508493, −24.227578506583342377768588016771, −23.905948562043300715355054010818, −22.76226914069882804662533123780, −21.5376453609502772251234900211, −20.62507370688710274437015699528, −19.66453172014886800672105259835, −18.7435654075531251978722730676, −18.140587791201925561925162275965, −17.07668003658712640493257494934, −16.15280939610208828641885417927, −15.61611618364604930755290916570, −14.33398909817620666445084908318, −13.6225297583293317506671293006, −12.165066789940219766701541052857, −11.01763111133964523185804818477, −10.3818277358323394893524794739, −9.12734112474137457243580930261, −8.38414038595953087707868777250, −7.436561213281500047025447454999, −6.21460585479946598545528472629, −5.53374884630856404575103357185, −3.92206230258341180708049652707, −2.39198105194598965822236697723, −0.990641260939589867970603068997, 1.04521065967180592813459988990, 2.41933484969459541386383337417, 3.4628785297976538996232545406, 4.84982629675761368904988875422, 6.32020441162856063722698442172, 7.50466562963151933380204924001, 8.199784143344322094360587280514, 9.48276543069226649283812685420, 10.09158078957851704953684382824, 11.21709421382335595533716985244, 12.02942530005477724807569146422, 13.02540143936131677117190767775, 14.023799190355879434160696990231, 15.70732197393685237980746994900, 15.90650884985575300362610918301, 17.36316941154106795517176732060, 18.007435305894725011158325694073, 18.69857739582665082780336370001, 19.91852122992349142954715700661, 20.48595775924034286407539411197, 21.28410106294143067047826472038, 22.42967977102055343665311804052, 23.22158803389177511015870872418, 24.56602635785521400516018031676, 25.36491202587493997696387379265

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