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Properties

Label 2-2352-588.299-c0-0-0
Degree $2$
Conductor $2352$
Sign $0.995 + 0.0960i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.826 − 0.563i)3-s + (−0.988 + 0.149i)7-s + (0.365 + 0.930i)9-s + (−0.880 + 0.702i)13-s + (0.988 − 1.71i)19-s + (0.900 + 0.433i)21-s + (0.988 + 0.149i)25-s + (0.222 − 0.974i)27-s + (0.955 + 1.65i)31-s + (1.40 + 0.432i)37-s + (1.12 − 0.0841i)39-s + (−0.807 + 1.67i)43-s + (0.955 − 0.294i)49-s + (−1.78 + 0.858i)57-s + (0.574 − 1.86i)61-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)3-s + (−0.988 + 0.149i)7-s + (0.365 + 0.930i)9-s + (−0.880 + 0.702i)13-s + (0.988 − 1.71i)19-s + (0.900 + 0.433i)21-s + (0.988 + 0.149i)25-s + (0.222 − 0.974i)27-s + (0.955 + 1.65i)31-s + (1.40 + 0.432i)37-s + (1.12 − 0.0841i)39-s + (−0.807 + 1.67i)43-s + (0.955 − 0.294i)49-s + (−1.78 + 0.858i)57-s + (0.574 − 1.86i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.995 + 0.0960i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (2063, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.995 + 0.0960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7595442347\)
\(L(\frac12)\) \(\approx\) \(0.7595442347\)
\(L(1)\) 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 \( 1 + (0.826 + 0.563i)T \)
7 \( 1 + (0.988 - 0.149i)T \)
good5 \( 1 + (-0.988 - 0.149i)T^{2} \)
11 \( 1 + (-0.733 - 0.680i)T^{2} \)
13 \( 1 + (0.880 - 0.702i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (0.0747 - 0.997i)T^{2} \)
19 \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.0747 + 0.997i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (-0.955 - 1.65i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.40 - 0.432i)T + (0.826 + 0.563i)T^{2} \)
41 \( 1 + (0.623 - 0.781i)T^{2} \)
43 \( 1 + (0.807 - 1.67i)T + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.955 + 0.294i)T^{2} \)
53 \( 1 + (-0.826 + 0.563i)T^{2} \)
59 \( 1 + (0.988 - 0.149i)T^{2} \)
61 \( 1 + (-0.574 + 1.86i)T + (-0.826 - 0.563i)T^{2} \)
67 \( 1 + (-1.72 + 0.997i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.202 + 1.34i)T + (-0.955 - 0.294i)T^{2} \)
79 \( 1 + (-0.510 - 0.294i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (-0.733 + 0.680i)T^{2} \)
97 \( 1 - 1.56iT - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.408542692245525696746936584255, −8.314889041130287045363356297821, −7.37049933630393438657065583942, −6.66309546982940598221639349508, −6.38908037503390459851290135046, −4.98633409821035557964106593449, −4.82576148661111299605826956834, −3.22636396188338260985733416851, −2.41634162783821338818507428477, −0.932992812627650320666213126638, 0.797760276661416708732099727987, 2.61773302552473009506142427662, 3.59807986642920122188345303001, 4.33289222913873355711232934996, 5.46680708748877349089775792529, 5.84773210437574806276226188166, 6.80180283424385081448573896039, 7.50262168408493847345531720056, 8.452690790526729040863061030986, 9.561434027659627218937433855554

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