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Properties

Label 1-315-315.214-r0-0-0
Degree $1$
Conductor $315$
Sign $0.690 + 0.723i$
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  − 2-s + 4-s − 8-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + 16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)26-s + (−0.5 − 0.866i)29-s + 31-s − 32-s + (−0.5 − 0.866i)34-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + 16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)26-s + (−0.5 − 0.866i)29-s + 31-s − 32-s + (−0.5 − 0.866i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.690 + 0.723i$
Analytic conductor: \(1.46285\)
Root analytic conductor: \(1.46285\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 315,\ (0:\ ),\ 0.690 + 0.723i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6973718801 + 0.2986697347i\)
\(L(\frac12)\) \(\approx\) \(0.6973718801 + 0.2986697347i\)
\(L(1)\) \(\approx\) \(0.7029183920 + 0.09803029914i\)
\(L(1)\) \(\approx\) \(0.7029183920 + 0.09803029914i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 - T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−25.319001305719427082774941010183, −24.20859291777744372181710188122, −23.68514645958731235377312194356, −22.29147558772776199547511773066, −21.14426701226040714744321317971, −20.67460412262162674441488346700, −19.41157211809700430417504929981, −18.7711310573133294513998537600, −18.03782629244256684633952184605, −16.84808135621073050280771343136, −16.26820774450642618943666644126, −15.351145973389531627470283925234, −14.20138689695441192490971479911, −13.10147121777073921547730930189, −11.805629221400044387895011064065, −11.08722866457559694462186656456, −10.159985882006666804623227335132, −9.006168022886328707305385176198, −8.389141049327466856274372135658, −7.1487826598589263342768689799, −6.32773569739068958959672955195, −5.03424675974209429754510994361, −3.3653104673916634490882894604, −2.27598636880795436964486003322, −0.76348293408459853018640275375, 1.26269482897681912002713583224, 2.49378609777426484246325922101, 3.785623670593405827404168169328, 5.465240868548750617601626380, 6.42353740328047845559127146181, 7.72659664903244246441934431635, 8.21836806469695458046528723929, 9.58472264639141652965827006537, 10.267187057789013548555621856593, 11.18788637502465448177593477424, 12.3316049515598151367922958925, 13.15592029691364929678535970461, 14.79647798096610300895054318461, 15.35875902337042038013203272938, 16.40454493521745701126861629386, 17.361685291752814900785060177705, 18.01699587392931080882213117893, 18.99588101375694693411737268198, 19.7775722881571833693907245956, 20.80725477095978614301776335855, 21.31051821421897942482612456666, 22.86442779173130581207012818204, 23.532838623147381235164524655968, 24.782641698682895410725119992685, 25.42651141626657500239798670027

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