Properties

Label 2-1407-1407.23-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.967 - 0.251i$
Analytic cond. $0.702184$
Root an. cond. $0.837964$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.928 − 0.371i)3-s + (0.841 + 0.540i)4-s + (−0.327 + 0.945i)7-s + (0.723 − 0.690i)9-s + (0.981 + 0.189i)12-s + (−1.88 − 0.363i)13-s + (0.415 + 0.909i)16-s + (1.50 − 0.442i)19-s + (0.0475 + 0.998i)21-s + (0.981 + 0.189i)25-s + (0.415 − 0.909i)27-s + (−0.786 + 0.618i)28-s + (−0.544 − 0.627i)31-s + (0.981 − 0.189i)36-s − 0.654·37-s + ⋯
L(s)  = 1  + (0.928 − 0.371i)3-s + (0.841 + 0.540i)4-s + (−0.327 + 0.945i)7-s + (0.723 − 0.690i)9-s + (0.981 + 0.189i)12-s + (−1.88 − 0.363i)13-s + (0.415 + 0.909i)16-s + (1.50 − 0.442i)19-s + (0.0475 + 0.998i)21-s + (0.981 + 0.189i)25-s + (0.415 − 0.909i)27-s + (−0.786 + 0.618i)28-s + (−0.544 − 0.627i)31-s + (0.981 − 0.189i)36-s − 0.654·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.967 - 0.251i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.967 - 0.251i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.674777733\)
\(L(\frac12)\) \(\approx\) \(1.674777733\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.928 + 0.371i)T \)
7 \( 1 + (0.327 - 0.945i)T \)
67 \( 1 + (0.327 + 0.945i)T \)
good2 \( 1 + (-0.841 - 0.540i)T^{2} \)
5 \( 1 + (-0.981 - 0.189i)T^{2} \)
11 \( 1 + (0.654 - 0.755i)T^{2} \)
13 \( 1 + (1.88 + 0.363i)T + (0.928 + 0.371i)T^{2} \)
17 \( 1 + (0.995 - 0.0950i)T^{2} \)
19 \( 1 + (-1.50 + 0.442i)T + (0.841 - 0.540i)T^{2} \)
23 \( 1 + (0.959 + 0.281i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
37 \( 1 + 0.654T + T^{2} \)
41 \( 1 + (0.995 - 0.0950i)T^{2} \)
43 \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 + (-0.723 + 0.690i)T^{2} \)
53 \( 1 + (0.995 + 0.0950i)T^{2} \)
59 \( 1 + (0.786 + 0.618i)T^{2} \)
61 \( 1 + (0.653 - 1.43i)T + (-0.654 - 0.755i)T^{2} \)
71 \( 1 + (-0.580 + 0.814i)T^{2} \)
73 \( 1 + (1.15 + 1.62i)T + (-0.327 + 0.945i)T^{2} \)
79 \( 1 + (0.642 + 0.123i)T + (0.928 + 0.371i)T^{2} \)
83 \( 1 + (0.327 + 0.945i)T^{2} \)
89 \( 1 + (-0.723 - 0.690i)T^{2} \)
97 \( 1 + (0.928 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.591488507201809707903616747246, −8.993188355423891988056438216899, −8.045629328939717679487856717304, −7.35539547125313896453366203587, −6.88376969033395920596786031013, −5.74881683777848139250449080899, −4.71734281141685859263710553239, −3.15938560766989107371438151341, −2.86511618599689854562210566066, −1.83980521490681230409088261985, 1.52246922409557189754002494236, 2.68701209034255929600578114082, 3.45792034924537326809677821364, 4.68922681380966415017445012809, 5.38675228255060707376411559370, 6.95199229544241302733438700383, 7.11673532636789034541500952286, 7.943971312054390655419046785159, 9.128885935506702226511635979779, 9.965767547055557872681611970765

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