Properties

Label 2-1300-65.64-c1-0-13
Degree $2$
Conductor $1300$
Sign $0.961 + 0.275i$
Analytic cond. $10.3805$
Root an. cond. $3.22188$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.339i·3-s + 3.88·7-s + 2.88·9-s − 1.54i·11-s + (−0.660 − 3.54i)13-s + 2.86i·19-s + 1.32i·21-s − 5.42i·23-s + 2i·27-s + 5.20·29-s − 6.22i·31-s + 0.524·33-s − 8.56·37-s + (1.20 − 0.224i)39-s + 9.08i·41-s + ⋯
L(s)  = 1  + 0.196i·3-s + 1.46·7-s + 0.961·9-s − 0.465i·11-s + (−0.183 − 0.983i)13-s + 0.657i·19-s + 0.288i·21-s − 1.13i·23-s + 0.384i·27-s + 0.966·29-s − 1.11i·31-s + 0.0913·33-s − 1.40·37-s + (0.192 − 0.0359i)39-s + 1.41i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.961 + 0.275i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.961 + 0.275i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.105990378\)
\(L(\frac12)\) \(\approx\) \(2.105990378\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 + (0.660 + 3.54i)T \)
good3 \( 1 - 0.339iT - 3T^{2} \)
7 \( 1 - 3.88T + 7T^{2} \)
11 \( 1 + 1.54iT - 11T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 2.86iT - 19T^{2} \)
23 \( 1 + 5.42iT - 23T^{2} \)
29 \( 1 - 5.20T + 29T^{2} \)
31 \( 1 + 6.22iT - 31T^{2} \)
37 \( 1 + 8.56T + 37T^{2} \)
41 \( 1 - 9.08iT - 41T^{2} \)
43 \( 1 + 0.980iT - 43T^{2} \)
47 \( 1 + 6.52T + 47T^{2} \)
53 \( 1 + 6.44iT - 53T^{2} \)
59 \( 1 - 4.45iT - 59T^{2} \)
61 \( 1 - 9.65T + 61T^{2} \)
67 \( 1 - 6.97T + 67T^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 - 3.43T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + 8.56T + 83T^{2} \)
89 \( 1 + 17.1iT - 89T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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

−9.850830263715651207515544811593, −8.494108694458214852886665469711, −8.174059971775701215755482900113, −7.30108202350398069223513215236, −6.28255240534509867243023849184, −5.20066851066008326175259463985, −4.61453352908165580185550950457, −3.59827462987004916157621817741, −2.24704616834037225355164745271, −1.05399794786229604523363617308, 1.38700541683735393516857829277, 2.11677515441819475717118098002, 3.73825496080945422225693934581, 4.73100533158260182350020353396, 5.17802971479965389165005469833, 6.68036515383072158708121525818, 7.18961004544745033772543716226, 8.011939750472846202920500540259, 8.846361860490412467081635522100, 9.652833345869602705329307863661

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