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Properties

Label 2-315-63.20-c1-0-25
Degree $2$
Conductor $315$
Sign $-0.223 + 0.974i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.552 + 0.319i)2-s + (−0.662 + 1.60i)3-s + (−0.796 − 1.37i)4-s + (−0.5 − 0.866i)5-s + (−0.877 + 0.672i)6-s + (−1.77 − 1.96i)7-s − 2.29i·8-s + (−2.12 − 2.12i)9-s − 0.638i·10-s + (−2.22 − 1.28i)11-s + (2.73 − 0.359i)12-s + (−2.91 + 1.68i)13-s + (−0.353 − 1.65i)14-s + (1.71 − 0.226i)15-s + (−0.860 + 1.49i)16-s + 2.60·17-s + ⋯
L(s)  = 1  + (0.390 + 0.225i)2-s + (−0.382 + 0.923i)3-s + (−0.398 − 0.689i)4-s + (−0.223 − 0.387i)5-s + (−0.358 + 0.274i)6-s + (−0.670 − 0.742i)7-s − 0.810i·8-s + (−0.707 − 0.707i)9-s − 0.201i·10-s + (−0.671 − 0.387i)11-s + (0.789 − 0.103i)12-s + (−0.807 + 0.466i)13-s + (−0.0944 − 0.441i)14-s + (0.443 − 0.0583i)15-s + (−0.215 + 0.372i)16-s + 0.632·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.223 + 0.974i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.223 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.395024 - 0.495989i\)
\(L(\frac12)\) \(\approx\) \(0.395024 - 0.495989i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.662 - 1.60i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (1.77 + 1.96i)T \)
good2 \( 1 + (-0.552 - 0.319i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (2.22 + 1.28i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.91 - 1.68i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.60T + 17T^{2} \)
19 \( 1 + 3.95iT - 19T^{2} \)
23 \( 1 + (-4.63 + 2.67i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.59 + 1.50i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.64 - 3.26i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.47T + 37T^{2} \)
41 \( 1 + (2.74 + 4.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.78 - 8.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.53 + 9.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.57iT - 53T^{2} \)
59 \( 1 + (4.28 + 7.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.81 + 2.20i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.92 + 8.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.71iT - 71T^{2} \)
73 \( 1 - 6.51iT - 73T^{2} \)
79 \( 1 + (1.07 - 1.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.62 + 6.28i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + (0.235 + 0.136i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.15215559237867745074495140968, −10.42996342846621856744821618254, −9.596964189901215347822182354242, −8.933841635946294264508561046337, −7.34092952767051773823351563879, −6.23879830840259877071597119488, −5.15407634969725837722037434845, −4.49331719339297258420012279805, −3.30055558728072535354649188208, −0.41008130297162975931925027995, 2.40167386431401754442630419995, 3.35484791058124487601919039622, 5.06864585748392931481300078307, 5.89617546714233810534360707129, 7.34255755574860349197348907995, 7.80801391745441087163590977422, 9.007691425870077730155399159470, 10.18130958469031948252098145563, 11.39829779762250281110502056461, 12.11913004177826669393472383231

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