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Properties

Label 2-315-7.6-c2-0-3
Degree $2$
Conductor $315$
Sign $-0.879 - 0.476i$
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  + 1.71·2-s − 1.06·4-s − 2.23i·5-s + (−3.33 + 6.15i)7-s − 8.67·8-s − 3.82i·10-s − 17.0·11-s + 16.3i·13-s + (−5.70 + 10.5i)14-s − 10.5·16-s − 13.4i·17-s + 13.7i·19-s + 2.38i·20-s − 29.1·22-s + 16.6·23-s + ⋯
L(s)  = 1  + 0.856·2-s − 0.267·4-s − 0.447i·5-s + (−0.476 + 0.879i)7-s − 1.08·8-s − 0.382i·10-s − 1.54·11-s + 1.25i·13-s + (−0.407 + 0.752i)14-s − 0.661·16-s − 0.789i·17-s + 0.723i·19-s + 0.119i·20-s − 1.32·22-s + 0.722·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.476i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.879 - 0.476i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ -0.879 - 0.476i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.136170 + 0.537392i\)
\(L(\frac12)\) \(\approx\) \(0.136170 + 0.537392i\)
\(L(2)\) 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 \( 1 + 2.23iT \)
7 \( 1 + (3.33 - 6.15i)T \)
good2 \( 1 - 1.71T + 4T^{2} \)
11 \( 1 + 17.0T + 121T^{2} \)
13 \( 1 - 16.3iT - 169T^{2} \)
17 \( 1 + 13.4iT - 289T^{2} \)
19 \( 1 - 13.7iT - 361T^{2} \)
23 \( 1 - 16.6T + 529T^{2} \)
29 \( 1 + 32.1T + 841T^{2} \)
31 \( 1 + 6.74iT - 961T^{2} \)
37 \( 1 + 69.2T + 1.36e3T^{2} \)
41 \( 1 - 39.7iT - 1.68e3T^{2} \)
43 \( 1 - 43.2T + 1.84e3T^{2} \)
47 \( 1 + 40.1iT - 2.20e3T^{2} \)
53 \( 1 + 22.5T + 2.80e3T^{2} \)
59 \( 1 + 81.6iT - 3.48e3T^{2} \)
61 \( 1 - 14.9iT - 3.72e3T^{2} \)
67 \( 1 - 72.0T + 4.48e3T^{2} \)
71 \( 1 - 25.7T + 5.04e3T^{2} \)
73 \( 1 - 75.0iT - 5.32e3T^{2} \)
79 \( 1 - 80.0T + 6.24e3T^{2} \)
83 \( 1 - 102. iT - 6.88e3T^{2} \)
89 \( 1 - 128. iT - 7.92e3T^{2} \)
97 \( 1 + 159. iT - 9.40e3T^{2} \)
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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

−12.15276290414611381511468106619, −11.18323867049889214945686579986, −9.750650340940215623263683048314, −9.109346182998546328979723421553, −8.147728519603837749057944933822, −6.74847494128903292048784919410, −5.50602616021929673171182116372, −5.00454739454030426225163768687, −3.64800892905279357806499302171, −2.38836299672036345281508402002, 0.18913789643129342347496944918, 2.83376336762294145578989315998, 3.69472873214064784907391160768, 5.00843141659882115361178682835, 5.81922694569382529183522723071, 7.09344643809479272765894249913, 8.023004610382337366027583930534, 9.243720030048426224643602854824, 10.48272731213836354646761746464, 10.78267535719253438096885011308

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