↑

Properties

Label 2-315-21.17-c3-0-18
Degree $2$
Conductor $315$
Sign $0.931 + 0.362i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.57 + 0.908i)2-s + (−2.34 + 4.06i)4-s + (2.5 + 4.33i)5-s + (−16.6 + 8.17i)7-s − 23.0i·8-s + (−7.86 − 4.54i)10-s + (−9.99 − 5.76i)11-s − 89.2i·13-s + (18.7 − 27.9i)14-s + (2.18 + 3.77i)16-s + (−59.0 + 102. i)17-s + (98.7 − 57.0i)19-s − 23.4·20-s + 20.9·22-s + (19.6 − 11.3i)23-s + ⋯
L(s)  = 1  + (−0.556 + 0.321i)2-s + (−0.293 + 0.508i)4-s + (0.223 + 0.387i)5-s + (−0.897 + 0.441i)7-s − 1.01i·8-s + (−0.248 − 0.143i)10-s + (−0.273 − 0.158i)11-s − 1.90i·13-s + (0.357 − 0.533i)14-s + (0.0340 + 0.0590i)16-s + (−0.842 + 1.45i)17-s + (1.19 − 0.688i)19-s − 0.262·20-s + 0.203·22-s + (0.178 − 0.102i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.362i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.931 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.931 + 0.362i$
Analytic conductor: \(18.5856\)
Root analytic conductor: \(4.31110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :3/2),\ 0.931 + 0.362i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7875639448\)
\(L(\frac12)\) \(\approx\) \(0.7875639448\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (16.6 - 8.17i)T \)
good2 \( 1 + (1.57 - 0.908i)T + (4 - 6.92i)T^{2} \)
11 \( 1 + (9.99 + 5.76i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 89.2iT - 2.19e3T^{2} \)
17 \( 1 + (59.0 - 102. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-98.7 + 57.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-19.6 + 11.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 38.6iT - 2.43e4T^{2} \)
31 \( 1 + (12.6 + 7.27i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (43.4 + 75.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 297.T + 6.89e4T^{2} \)
43 \( 1 - 116.T + 7.95e4T^{2} \)
47 \( 1 + (162. + 282. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-322. - 186. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-2.56 + 4.43i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-490. + 283. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-418. + 724. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 8.15iT - 3.57e5T^{2} \)
73 \( 1 + (887. + 512. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-359. - 622. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.33e3T + 5.71e5T^{2} \)
89 \( 1 + (135. + 234. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 947. iT - 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.89619013567922077844733619330, −10.11014716150180124964585270400, −9.216940194910018011721969185587, −8.342845975797893946185844156762, −7.45342467938329904573457420611, −6.41116267807934182893189501440, −5.39313557460582529858494771487, −3.67033699251266958616786604274, −2.77249402950576662655674980647, −0.43929481737030063423711692203, 1.00405824716594685703069400165, 2.42541907277066398666743885266, 4.17739016797849657583796566270, 5.22960565337706758004106381627, 6.46170361280258763088435426880, 7.44885986064944321528814645330, 8.933144896057859845711416081130, 9.437840330087471923938020088374, 10.05638115602703872732235424208, 11.25679133064439194869896563564

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