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Properties

Label 2-315-5.2-c2-0-25
Degree $2$
Conductor $315$
Sign $0.202 - 0.979i$
Analytic cond. $8.58312$
Root an. cond. $2.92969$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.08 − 2.08i)2-s + 4.73i·4-s + (−0.137 − 4.99i)5-s + (−1.87 − 1.87i)7-s + (1.53 − 1.53i)8-s + (−10.1 + 10.7i)10-s − 2.70·11-s + (−2.37 + 2.37i)13-s + 7.81i·14-s + 12.5·16-s + (−16.3 − 16.3i)17-s − 9.18i·19-s + (23.6 − 0.649i)20-s + (5.64 + 5.64i)22-s + (−21.4 + 21.4i)23-s + ⋯
L(s)  = 1  + (−1.04 − 1.04i)2-s + 1.18i·4-s + (−0.0274 − 0.999i)5-s + (−0.267 − 0.267i)7-s + (0.191 − 0.191i)8-s + (−1.01 + 1.07i)10-s − 0.245·11-s + (−0.183 + 0.183i)13-s + 0.558i·14-s + 0.782·16-s + (−0.963 − 0.963i)17-s − 0.483i·19-s + (1.18 − 0.0324i)20-s + (0.256 + 0.256i)22-s + (−0.932 + 0.932i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.202 - 0.979i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ 0.202 - 0.979i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0269267 + 0.0219176i\)
\(L(\frac12)\) \(\approx\) \(0.0269267 + 0.0219176i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.137 + 4.99i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good2 \( 1 + (2.08 + 2.08i)T + 4iT^{2} \)
11 \( 1 + 2.70T + 121T^{2} \)
13 \( 1 + (2.37 - 2.37i)T - 169iT^{2} \)
17 \( 1 + (16.3 + 16.3i)T + 289iT^{2} \)
19 \( 1 + 9.18iT - 361T^{2} \)
23 \( 1 + (21.4 - 21.4i)T - 529iT^{2} \)
29 \( 1 - 52.3iT - 841T^{2} \)
31 \( 1 + 5.01T + 961T^{2} \)
37 \( 1 + (-23.2 - 23.2i)T + 1.36e3iT^{2} \)
41 \( 1 - 60.5T + 1.68e3T^{2} \)
43 \( 1 + (8.78 - 8.78i)T - 1.84e3iT^{2} \)
47 \( 1 + (2.24 + 2.24i)T + 2.20e3iT^{2} \)
53 \( 1 + (-25.6 + 25.6i)T - 2.80e3iT^{2} \)
59 \( 1 - 100. iT - 3.48e3T^{2} \)
61 \( 1 + 82.1T + 3.72e3T^{2} \)
67 \( 1 + (65.1 + 65.1i)T + 4.48e3iT^{2} \)
71 \( 1 - 22.8T + 5.04e3T^{2} \)
73 \( 1 + (5.38 - 5.38i)T - 5.32e3iT^{2} \)
79 \( 1 - 117. iT - 6.24e3T^{2} \)
83 \( 1 + (85.5 - 85.5i)T - 6.88e3iT^{2} \)
89 \( 1 + 119. iT - 7.92e3T^{2} \)
97 \( 1 + (55.1 + 55.1i)T + 9.40e3iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.70075426700684905534670938794, −9.587052778113531437990461517389, −9.167036636650116527691358024036, −8.221776878405126657126821081975, −7.17348138682535905905089868168, −5.57168601520163898526394899807, −4.32433456769907039550751301525, −2.81704238438118240782423541759, −1.41875694949684677846391194626, −0.02362835841349120692011332710, 2.39664210826052631645385571014, 4.01744104294645452345796229371, 5.97760976662915448505051080333, 6.35974139912342358813072664200, 7.55078624509756233115266067561, 8.169436292215725104775094126964, 9.264703468794996500728430287010, 10.13529227550120066984113863892, 10.82464534564487061336271078504, 12.06437247627456901682692757739

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