↑

Properties

Label 2-2352-7.2-c1-0-20
Degree $2$
Conductor $2352$
Sign $0.900 - 0.435i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
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  + (0.5 + 0.866i)3-s + (1.70 − 2.95i)5-s + (−0.499 + 0.866i)9-s + (0.414 + 0.717i)11-s + 4.24·13-s + 3.41·15-s + (3.70 + 6.42i)17-s + (−3.41 + 5.91i)19-s + (−2.41 + 4.18i)23-s + (−3.32 − 5.76i)25-s − 0.999·27-s + 2.82·29-s + (1.41 + 2.44i)31-s + (−0.414 + 0.717i)33-s + (0.828 − 1.43i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.763 − 1.32i)5-s + (−0.166 + 0.288i)9-s + (0.124 + 0.216i)11-s + 1.17·13-s + 0.881·15-s + (0.899 + 1.55i)17-s + (−0.783 + 1.35i)19-s + (−0.503 + 0.871i)23-s + (−0.665 − 1.15i)25-s − 0.192·27-s + 0.525·29-s + (0.254 + 0.439i)31-s + (−0.0721 + 0.124i)33-s + (0.136 − 0.235i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.900 - 0.435i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.448605419\)
\(L(\frac12)\) \(\approx\) \(2.448605419\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1.70 + 2.95i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.414 - 0.717i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + (-3.70 - 6.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.41 - 5.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.41 - 4.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + (-1.41 - 2.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.828 + 1.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 + (-2.24 + 3.88i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.24 + 7.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.53 + 9.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.65 - 9.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + (-3.87 - 6.71i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.82 + 11.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-2.87 + 4.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.242T + 97T^{2} \)
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

−8.945615313439537673088395087414, −8.359968547567570279542147578064, −7.945553334682087690711512648223, −6.42104170246016069615934109289, −5.79611949242956856891820242829, −5.18631134901648262582703086394, −4.05471275423980871607708193926, −3.61541654655127987695948185593, −1.94893708431035641060370700923, −1.27913780507361928967728227294, 0.907955569593034552632560229170, 2.38607773969250771241052319631, 2.82178667572197851195289853302, 3.85056203656841762425579502241, 5.09952121097565407255483421707, 6.13926056678923385846675956234, 6.56643607814344306001758065622, 7.23207398254984314235968548253, 8.119924615522079928069944269877, 8.957983537579882401489386707920

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