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Properties

Label 1-315-315.299-r0-0-0
Degree $1$
Conductor $315$
Sign $-0.220 + 0.975i$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + 8-s − 11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)22-s + 23-s + (−0.5 − 0.866i)26-s + (0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s + (−0.5 − 0.866i)32-s + (0.5 + 0.866i)34-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + 8-s − 11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)22-s + 23-s + (−0.5 − 0.866i)26-s + (0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s + (−0.5 − 0.866i)32-s + (0.5 + 0.866i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.220 + 0.975i$
Analytic conductor: \(1.46285\)
Root analytic conductor: \(1.46285\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 315,\ (0:\ ),\ -0.220 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4983018361 + 0.6235167866i\)
\(L(\frac12)\) \(\approx\) \(0.4983018361 + 0.6235167866i\)
\(L(1)\) \(\approx\) \(0.6778353554 + 0.3545981715i\)
\(L(1)\) \(\approx\) \(0.6778353554 + 0.3545981715i\)

Euler product

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

−25.182064387693999606173539889947, −23.98552872662073170022832459509, −22.94233685749476313181088282716, −22.128500596137465366706968154597, −21.15303407055539617635163594257, −20.52864822373084577221807494128, −19.47363208351506646782574897290, −18.829398362367168807505114774143, −17.717907452076154066034082062436, −17.19627508130316251200741108947, −15.948871227751219295643795140304, −14.980818747726047364443360163173, −13.56937791870664260521212594687, −12.88469832749511566592844277741, −11.97921222410750348728801074272, −10.8402952846735955817946105013, −10.201616643433827170623022679177, −9.16345340152258584212568797862, −8.05727106538134063356138040303, −7.34043826670450627768295982789, −5.621773002716690419991591422458, −4.522681351339846839182568360529, −3.17656671875672487385474191027, −2.320184489997965178170024874828, −0.69009698424963770738238969615, 1.287118108123940877353999640609, 2.91793371683570929103274552722, 4.65180811028656383855082695741, 5.37876343073717234285950437618, 6.65950476668699773478963699598, 7.49626743174118417391164030230, 8.437391267127029061574589490792, 9.53286721644353213858254544390, 10.276730373784183078940545461333, 11.46166556745500069489421196873, 12.7312284791751551504117884897, 13.88007171795030405612096559019, 14.54542027513993822673353500554, 15.67183792854491854965712922250, 16.36983322806569116845680701134, 17.19609166409849467943372786511, 18.361516514193468614249739724970, 18.77213197764178460678830392133, 19.916249474069410849400703324334, 20.948057925294182338687327911961, 22.020685048449581012901345717197, 23.21787706958411084176032091157, 23.59438004425161227596138819855, 24.80811940301579846177124005280, 25.29916399276589510839822929417

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