Properties

Label 2-9295-1.1-c1-0-290
Degree $2$
Conductor $9295$
Sign $1$
Analytic cond. $74.2209$
Root an. cond. $8.61515$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.414·2-s + 2.82·3-s − 1.82·4-s + 5-s + 1.17·6-s + 2·7-s − 1.58·8-s + 5.00·9-s + 0.414·10-s − 11-s − 5.17·12-s + 0.828·14-s + 2.82·15-s + 3·16-s + 1.17·17-s + 2.07·18-s − 1.82·20-s + 5.65·21-s − 0.414·22-s + 2.82·23-s − 4.48·24-s + 25-s + 5.65·27-s − 3.65·28-s + 7.65·29-s + 1.17·30-s + 4.41·32-s + ⋯
L(s)  = 1  + 0.292·2-s + 1.63·3-s − 0.914·4-s + 0.447·5-s + 0.478·6-s + 0.755·7-s − 0.560·8-s + 1.66·9-s + 0.130·10-s − 0.301·11-s − 1.49·12-s + 0.221·14-s + 0.730·15-s + 0.750·16-s + 0.284·17-s + 0.488·18-s − 0.408·20-s + 1.23·21-s − 0.0883·22-s + 0.589·23-s − 0.915·24-s + 0.200·25-s + 1.08·27-s − 0.691·28-s + 1.42·29-s + 0.213·30-s + 0.780·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9295 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9295\)    =    \(5 \cdot 11 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(74.2209\)
Root analytic conductor: \(8.61515\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{9295} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9295,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.562830744\)
\(L(\frac12)\) \(\approx\) \(4.562830744\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 \)
good2 \( 1 - 0.414T + 2T^{2} \)
3 \( 1 - 2.82T + 3T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
17 \( 1 - 1.17T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 - 7.65T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 3.65T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 0.343T + 53T^{2} \)
59 \( 1 - 9.65T + 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 - 4.48T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 6.82T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + 9.31T + 89T^{2} \)
97 \( 1 - 7.65T + 97T^{2} \)
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

−8.072030087044854451712609832327, −7.17228407899312398049856832382, −6.45363616445473831613587675061, −5.22545348697061647241077523690, −5.01938373227901756208932977384, −4.05630227843943496546017351971, −3.45003488722167814650762971944, −2.72503746672991041715810033517, −1.93781508675787687574555098567, −0.952888085064269769201358610919, 0.952888085064269769201358610919, 1.93781508675787687574555098567, 2.72503746672991041715810033517, 3.45003488722167814650762971944, 4.05630227843943496546017351971, 5.01938373227901756208932977384, 5.22545348697061647241077523690, 6.45363616445473831613587675061, 7.17228407899312398049856832382, 8.072030087044854451712609832327

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