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Properties

Label 2-48e2-9.5-c0-0-2
Degree $2$
Conductor $2304$
Sign $0.642 + 0.766i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
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  + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)9-s + (0.866 − 0.5i)11-s − 1.73i·17-s − 1.73·19-s + (−0.5 − 0.866i)25-s + 0.999i·27-s + (−0.499 + 0.866i)33-s + (1.5 + 0.866i)41-s + (−0.866 − 1.5i)43-s + (0.5 − 0.866i)49-s + (0.866 + 1.49i)51-s + (1.49 − 0.866i)57-s + (0.866 + 0.5i)59-s + (0.866 − 1.5i)67-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)9-s + (0.866 − 0.5i)11-s − 1.73i·17-s − 1.73·19-s + (−0.5 − 0.866i)25-s + 0.999i·27-s + (−0.499 + 0.866i)33-s + (1.5 + 0.866i)41-s + (−0.866 − 1.5i)43-s + (0.5 − 0.866i)49-s + (0.866 + 1.49i)51-s + (1.49 − 0.866i)57-s + (0.866 + 0.5i)59-s + (0.866 − 1.5i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7855029673\)
\(L(\frac12)\) \(\approx\) \(0.7855029673\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + 1.73iT - T^{2} \)
19 \( 1 + 1.73T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.142788823588313909403929356844, −8.522300366056296978897073657418, −7.36904559278583920852394297104, −6.56525597863044695490943265602, −6.04816706974671542623903465504, −5.06547542834992258641519991966, −4.33892791892797289599897993784, −3.55154673152441964930575617037, −2.24748419402585637127423440804, −0.63896316386965826928805405582, 1.39631768484973857788478946766, 2.22869031404860420791018203108, 3.89162785488803733323717631581, 4.41230288828844598445970233498, 5.57293075494166658867912107934, 6.25981445554587548154095423450, 6.77632484815982570105601750998, 7.71543742759436111296991731352, 8.420042895051131358168858662398, 9.300295049492619524933036041812

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