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Properties

Label 2-315-5.4-c3-0-16
Degree $2$
Conductor $315$
Sign $0.317 - 0.948i$
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  + 5.44i·2-s − 21.6·4-s + (−10.6 − 3.55i)5-s − 7i·7-s − 74.3i·8-s + (19.3 − 57.7i)10-s − 60.4·11-s + 64.0i·13-s + 38.1·14-s + 231.·16-s − 37.7i·17-s + 134.·19-s + (229. + 76.9i)20-s − 329. i·22-s + 52.6i·23-s + ⋯
L(s)  = 1  + 1.92i·2-s − 2.70·4-s + (−0.948 − 0.317i)5-s − 0.377i·7-s − 3.28i·8-s + (0.611 − 1.82i)10-s − 1.65·11-s + 1.36i·13-s + 0.727·14-s + 3.62·16-s − 0.538i·17-s + 1.62·19-s + (2.56 + 0.860i)20-s − 3.19i·22-s + 0.477i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.317 - 0.948i$
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: \((2,\ 315,\ (\ :3/2),\ 0.317 - 0.948i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7864217683\)
\(L(\frac12)\) \(\approx\) \(0.7864217683\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.6 + 3.55i)T \)
7 \( 1 + 7iT \)
good2 \( 1 - 5.44iT - 8T^{2} \)
11 \( 1 + 60.4T + 1.33e3T^{2} \)
13 \( 1 - 64.0iT - 2.19e3T^{2} \)
17 \( 1 + 37.7iT - 4.91e3T^{2} \)
19 \( 1 - 134.T + 6.85e3T^{2} \)
23 \( 1 - 52.6iT - 1.21e4T^{2} \)
29 \( 1 - 165.T + 2.43e4T^{2} \)
31 \( 1 + 71.9T + 2.97e4T^{2} \)
37 \( 1 + 48.7iT - 5.06e4T^{2} \)
41 \( 1 - 10.5T + 6.89e4T^{2} \)
43 \( 1 + 425. iT - 7.95e4T^{2} \)
47 \( 1 + 249. iT - 1.03e5T^{2} \)
53 \( 1 + 544. iT - 1.48e5T^{2} \)
59 \( 1 + 567.T + 2.05e5T^{2} \)
61 \( 1 - 614.T + 2.26e5T^{2} \)
67 \( 1 + 201. iT - 3.00e5T^{2} \)
71 \( 1 - 525.T + 3.57e5T^{2} \)
73 \( 1 - 137. iT - 3.89e5T^{2} \)
79 \( 1 - 234.T + 4.93e5T^{2} \)
83 \( 1 + 10.3iT - 5.71e5T^{2} \)
89 \( 1 + 20.9T + 7.04e5T^{2} \)
97 \( 1 + 1.38e3iT - 9.12e5T^{2} \)
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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

−11.53447603147248603092918309860, −10.08840933933028976362918648994, −9.126150812695611037851065771499, −8.205173124811788546801253638570, −7.44910080075171448517923686881, −6.89926959402309203126092440352, −5.38412738124917398076311169213, −4.78934401953088459294859633416, −3.59882593844713125282647112037, −0.44368623491182665429023565587, 0.828082418887787021486223412901, 2.73224652385644058939824761266, 3.19937916754547920080987648723, 4.62470241290740947646609205819, 5.52685048094048337535021268695, 7.81416721430229824164712152883, 8.256011698955554985262954300232, 9.577598795081644005056628439406, 10.50939938558480964736538679071, 10.91295878160994167131391103987

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