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Properties

Label 2-2352-7.6-c2-0-4
Degree $2$
Conductor $2352$
Sign $-0.654 - 0.755i$
Analytic cond. $64.0873$
Root an. cond. $8.00545$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s + 1.43i·5-s − 2.99·9-s − 6·11-s + 21.3i·13-s + 2.48·15-s − 8.95i·17-s − 7.22i·19-s + 37.4·23-s + 22.9·25-s + 5.19i·27-s − 33.9·29-s − 44.1i·31-s + 10.3i·33-s − 27.9·37-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.286i·5-s − 0.333·9-s − 0.545·11-s + 1.64i·13-s + 0.165·15-s − 0.526i·17-s − 0.380i·19-s + 1.62·23-s + 0.917·25-s + 0.192i·27-s − 1.17·29-s − 1.42i·31-s + 0.314i·33-s − 0.755·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.654 - 0.755i$
Analytic conductor: \(64.0873\)
Root analytic conductor: \(8.00545\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1),\ -0.654 - 0.755i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5883053295\)
\(L(\frac12)\) \(\approx\) \(0.5883053295\)
\(L(2)\) 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 \( 1 + 1.73iT \)
7 \( 1 \)
good5 \( 1 - 1.43iT - 25T^{2} \)
11 \( 1 + 6T + 121T^{2} \)
13 \( 1 - 21.3iT - 169T^{2} \)
17 \( 1 + 8.95iT - 289T^{2} \)
19 \( 1 + 7.22iT - 361T^{2} \)
23 \( 1 - 37.4T + 529T^{2} \)
29 \( 1 + 33.9T + 841T^{2} \)
31 \( 1 + 44.1iT - 961T^{2} \)
37 \( 1 + 27.9T + 1.36e3T^{2} \)
41 \( 1 - 54.8iT - 1.68e3T^{2} \)
43 \( 1 - 1.48T + 1.84e3T^{2} \)
47 \( 1 + 43.0iT - 2.20e3T^{2} \)
53 \( 1 + 85.4T + 2.80e3T^{2} \)
59 \( 1 - 41.2iT - 3.48e3T^{2} \)
61 \( 1 - 1.18iT - 3.72e3T^{2} \)
67 \( 1 + 4.39T + 4.48e3T^{2} \)
71 \( 1 + 137.T + 5.04e3T^{2} \)
73 \( 1 - 78.9iT - 5.32e3T^{2} \)
79 \( 1 + 98.3T + 6.24e3T^{2} \)
83 \( 1 - 110. iT - 6.88e3T^{2} \)
89 \( 1 - 20.7iT - 7.92e3T^{2} \)
97 \( 1 - 10.9iT - 9.40e3T^{2} \)
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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

−9.109987669758169076462712623196, −8.322451133546603021014104451770, −7.23625287217355732987791838208, −7.01757338984304940468336148489, −6.10780650640862366163714982280, −5.12921909690492362920680454393, −4.35911050047959865427445056936, −3.15450290194795871087312979783, −2.33944387392814605599131777472, −1.27101957041702755783803140690, 0.14452239273882443771199750249, 1.44553868120603611740888883098, 2.94731347048548606466473220012, 3.42183568417264908393510987487, 4.70743695271710049972874439783, 5.26798447902563472222446492105, 5.95273213458064814238941412921, 7.10539673644487213905905825476, 7.82468961959385188227944770791, 8.662064922897133674841048070154

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