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Properties

Label 2-315-5.4-c3-0-32
Degree $2$
Conductor $315$
Sign $0.420 + 0.907i$
Analytic cond. $18.5856$
Root an. cond. $4.31110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.33i·2-s − 3.14·4-s + (−10.1 + 4.70i)5-s − 7i·7-s + 16.2i·8-s + (−15.7 − 33.8i)10-s − 18.3·11-s + 10.1i·13-s + 23.3·14-s − 79.2·16-s − 24.6i·17-s − 77.4·19-s + (31.8 − 14.7i)20-s − 61.1i·22-s − 149. i·23-s + ⋯
L(s)  = 1  + 1.18i·2-s − 0.392·4-s + (−0.907 + 0.420i)5-s − 0.377i·7-s + 0.716i·8-s + (−0.496 − 1.07i)10-s − 0.502·11-s + 0.216i·13-s + 0.446·14-s − 1.23·16-s − 0.352i·17-s − 0.934·19-s + (0.356 − 0.165i)20-s − 0.592i·22-s − 1.35i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.420 + 0.907i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.420 + 0.907i$
Analytic conductor: \(18.5856\)
Root analytic conductor: \(4.31110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :3/2),\ 0.420 + 0.907i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1990086968\)
\(L(\frac12)\) \(\approx\) \(0.1990086968\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.1 - 4.70i)T \)
7 \( 1 + 7iT \)
good2 \( 1 - 3.33iT - 8T^{2} \)
11 \( 1 + 18.3T + 1.33e3T^{2} \)
13 \( 1 - 10.1iT - 2.19e3T^{2} \)
17 \( 1 + 24.6iT - 4.91e3T^{2} \)
19 \( 1 + 77.4T + 6.85e3T^{2} \)
23 \( 1 + 149. iT - 1.21e4T^{2} \)
29 \( 1 + 10.2T + 2.43e4T^{2} \)
31 \( 1 - 124.T + 2.97e4T^{2} \)
37 \( 1 + 215. iT - 5.06e4T^{2} \)
41 \( 1 + 495.T + 6.89e4T^{2} \)
43 \( 1 + 220. iT - 7.95e4T^{2} \)
47 \( 1 - 212. iT - 1.03e5T^{2} \)
53 \( 1 + 532. iT - 1.48e5T^{2} \)
59 \( 1 + 324.T + 2.05e5T^{2} \)
61 \( 1 - 653.T + 2.26e5T^{2} \)
67 \( 1 - 819. iT - 3.00e5T^{2} \)
71 \( 1 - 466.T + 3.57e5T^{2} \)
73 \( 1 - 173. iT - 3.89e5T^{2} \)
79 \( 1 + 810.T + 4.93e5T^{2} \)
83 \( 1 + 12.3iT - 5.71e5T^{2} \)
89 \( 1 + 33.8T + 7.04e5T^{2} \)
97 \( 1 - 810. iT - 9.12e5T^{2} \)
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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

−11.03573472489090387252773030583, −10.22918917274015430143816656480, −8.644133938464949396493755059822, −8.076600313338466681680390204063, −7.05037521063718376338967839162, −6.51817225016193940708071520046, −5.14355209845198245765867562418, −4.08976519044307939349352692728, −2.52514614682536391411721444323, −0.07116996621473162100066388571, 1.48511715141549567919687544593, 2.91286709688909924202383670501, 3.92131384523131566614649322539, 5.07223234542729756056883031043, 6.55697893390282166071706342221, 7.79444487118033990204329748773, 8.680254712316320382993198381204, 9.770805925649691822866413816977, 10.66013117119571284988676869726, 11.49419343858231062916400893927

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