↑

Properties

Label 2-2352-7.2-c1-0-23
Degree $2$
Conductor $2352$
Sign $-0.605 + 0.795i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (−2 + 3.46i)5-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)11-s − 6·13-s + 3.99·15-s + (2 + 3.46i)17-s + (−2 + 3.46i)19-s + (1 − 1.73i)23-s + (−5.49 − 9.52i)25-s + 0.999·27-s − 2·29-s + (0.999 − 1.73i)33-s + (−1 + 1.73i)37-s + (3 + 5.19i)39-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.894 + 1.54i)5-s + (−0.166 + 0.288i)9-s + (0.301 + 0.522i)11-s − 1.66·13-s + 1.03·15-s + (0.485 + 0.840i)17-s + (−0.458 + 0.794i)19-s + (0.208 − 0.361i)23-s + (−1.09 − 1.90i)25-s + 0.192·27-s − 0.371·29-s + (0.174 − 0.301i)33-s + (−0.164 + 0.284i)37-s + (0.480 + 0.832i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.605 + 0.795i$
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: \(1\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14T + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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

−8.480817509803977609972740628869, −7.55348933122337500701564719384, −7.33809046613458881967100321862, −6.55052557829311745954904751313, −5.79302421586826723913292357580, −4.60057245123585073529802858782, −3.75742431566922227731013445250, −2.79929930702269750869730196569, −1.92545138216769679377727655102, 0, 1.00992685978104315316509680473, 2.65178277640528781025229756983, 3.80820404799834546549040359708, 4.63280589077664668650350894496, 5.05088662128589329944514490018, 5.87336763868096134214583691192, 7.21172342600384768010233247166, 7.68056505602677513681760711610, 8.671451479643432302227765573359

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