↑

Properties

Label 2-2352-588.47-c0-0-0
Degree $2$
Conductor $2352$
Sign $-0.710 - 0.703i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
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  + (−0.365 + 0.930i)3-s + (0.955 + 0.294i)7-s + (−0.733 − 0.680i)9-s + (−1.81 + 0.414i)13-s + (−0.955 + 1.65i)19-s + (−0.623 + 0.781i)21-s + (−0.955 + 0.294i)25-s + (0.900 − 0.433i)27-s + (0.826 + 1.43i)31-s + (−0.123 + 0.0841i)37-s + (0.277 − 1.84i)39-s + (1.06 + 0.848i)43-s + (0.826 + 0.563i)49-s + (−1.19 − 1.49i)57-s + (−0.488 − 0.716i)61-s + ⋯
L(s)  = 1  + (−0.365 + 0.930i)3-s + (0.955 + 0.294i)7-s + (−0.733 − 0.680i)9-s + (−1.81 + 0.414i)13-s + (−0.955 + 1.65i)19-s + (−0.623 + 0.781i)21-s + (−0.955 + 0.294i)25-s + (0.900 − 0.433i)27-s + (0.826 + 1.43i)31-s + (−0.123 + 0.0841i)37-s + (0.277 − 1.84i)39-s + (1.06 + 0.848i)43-s + (0.826 + 0.563i)49-s + (−1.19 − 1.49i)57-s + (−0.488 − 0.716i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.710 - 0.703i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ -0.710 - 0.703i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7971787281\)
\(L(\frac12)\) \(\approx\) \(0.7971787281\)
\(L(1)\) 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 \( 1 + (0.365 - 0.930i)T \)
7 \( 1 + (-0.955 - 0.294i)T \)
good5 \( 1 + (0.955 - 0.294i)T^{2} \)
11 \( 1 + (0.0747 - 0.997i)T^{2} \)
13 \( 1 + (1.81 - 0.414i)T + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (-0.988 + 0.149i)T^{2} \)
19 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.988 - 0.149i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (-0.826 - 1.43i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.123 - 0.0841i)T + (0.365 - 0.930i)T^{2} \)
41 \( 1 + (-0.222 + 0.974i)T^{2} \)
43 \( 1 + (-1.06 - 0.848i)T + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.826 - 0.563i)T^{2} \)
53 \( 1 + (-0.365 - 0.930i)T^{2} \)
59 \( 1 + (-0.955 - 0.294i)T^{2} \)
61 \( 1 + (0.488 + 0.716i)T + (-0.365 + 0.930i)T^{2} \)
67 \( 1 + (0.258 - 0.149i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.587 - 1.90i)T + (-0.826 + 0.563i)T^{2} \)
79 \( 1 + (0.975 + 0.563i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.0747 + 0.997i)T^{2} \)
97 \( 1 + 1.94iT - T^{2} \)
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

−9.654574810179752226951648394491, −8.694419914255155747271358369283, −8.066514317368676415413399655840, −7.20196122926602774019326826830, −6.14248322533190565337309268140, −5.41142128842050564900850580703, −4.65344641654509430697688318084, −4.08579879840725750572968245492, −2.84453884871822355885456961829, −1.76611225388265193728008127396, 0.53751670027556986150188607865, 2.10122339311883474117057028454, 2.60797069367019901747665552432, 4.31547591593075501730557505863, 4.92758191098015681021947681814, 5.75671346969591066291819457612, 6.69739704460874496470738459959, 7.46356981583605707579972739684, 7.83888854813236219387059745636, 8.719339324842533233197522098459

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