↑

Properties

Label 2-2352-588.383-c0-0-1
Degree $2$
Conductor $2352$
Sign $0.964 + 0.263i$
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  + (0.988 + 0.149i)3-s + (−0.733 − 0.680i)7-s + (0.955 + 0.294i)9-s + (0.290 + 0.0663i)13-s + (0.733 − 1.26i)19-s + (−0.623 − 0.781i)21-s + (0.733 − 0.680i)25-s + (0.900 + 0.433i)27-s + (0.0747 + 0.129i)31-s + (−0.123 + 1.64i)37-s + (0.277 + 0.108i)39-s + (0.460 − 0.367i)43-s + (0.0747 + 0.997i)49-s + (0.914 − 1.14i)57-s + (0.865 + 0.0648i)61-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)3-s + (−0.733 − 0.680i)7-s + (0.955 + 0.294i)9-s + (0.290 + 0.0663i)13-s + (0.733 − 1.26i)19-s + (−0.623 − 0.781i)21-s + (0.733 − 0.680i)25-s + (0.900 + 0.433i)27-s + (0.0747 + 0.129i)31-s + (−0.123 + 1.64i)37-s + (0.277 + 0.108i)39-s + (0.460 − 0.367i)43-s + (0.0747 + 0.997i)49-s + (0.914 − 1.14i)57-s + (0.865 + 0.0648i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.964 + 0.263i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.964 + 0.263i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.643847421\)
\(L(\frac12)\) \(\approx\) \(1.643847421\)
\(L(1)\) 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 \( 1 + (-0.988 - 0.149i)T \)
7 \( 1 + (0.733 + 0.680i)T \)
good5 \( 1 + (-0.733 + 0.680i)T^{2} \)
11 \( 1 + (0.826 - 0.563i)T^{2} \)
13 \( 1 + (-0.290 - 0.0663i)T + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (0.365 + 0.930i)T^{2} \)
19 \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.365 - 0.930i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.0747 - 0.129i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.123 - 1.64i)T + (-0.988 - 0.149i)T^{2} \)
41 \( 1 + (-0.222 - 0.974i)T^{2} \)
43 \( 1 + (-0.460 + 0.367i)T + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.0747 - 0.997i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (0.733 + 0.680i)T^{2} \)
61 \( 1 + (-0.865 - 0.0648i)T + (0.988 + 0.149i)T^{2} \)
67 \( 1 + (1.61 - 0.930i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.766 + 0.825i)T + (-0.0747 + 0.997i)T^{2} \)
79 \( 1 + (1.72 + 0.997i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.826 + 0.563i)T^{2} \)
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

−9.023925532620602521851379545759, −8.555390206077132087005383209904, −7.52430712480935711679832467446, −7.00940335692655604554513305045, −6.23172271224150063000725911132, −4.93239624871125535220311533395, −4.22734072912939556744499049627, −3.25310131063809947687917087364, −2.66293103715532086770108652658, −1.18790844415912775965690583255, 1.46655236902268370341217876297, 2.59597194773686617918502438155, 3.36133478507066678888379074396, 4.11132117060967063445944615589, 5.38672421472206358756914138723, 6.10152942635774242872459128801, 7.05716004569121333196475656458, 7.69434078603212078127296471224, 8.563854759045929040749536631691, 9.118991241870959132304854908964

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