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Properties

Label 2-48e2-48.11-c1-0-22
Degree $2$
Conductor $2304$
Sign $0.845 + 0.533i$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.16 − 1.16i)5-s + 3.74·7-s + (1.64 + 1.64i)11-s + (0.645 − 0.645i)13-s − 6.57i·17-s + (1.41 + 1.41i)19-s − 6i·23-s + 2.29i·25-s + (5.40 + 5.40i)29-s − 0.913i·31-s + (4.35 − 4.35i)35-s + (−1 − i)37-s − 6.57·41-s + (3.74 − 3.74i)43-s + 6·47-s + ⋯
L(s)  = 1  + (0.520 − 0.520i)5-s + 1.41·7-s + (0.496 + 0.496i)11-s + (0.179 − 0.179i)13-s − 1.59i·17-s + (0.324 + 0.324i)19-s − 1.25i·23-s + 0.458i·25-s + (1.00 + 1.00i)29-s − 0.164i·31-s + (0.736 − 0.736i)35-s + (−0.164 − 0.164i)37-s − 1.02·41-s + (0.570 − 0.570i)43-s + 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.845 + 0.533i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.845 + 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.512465257\)
\(L(\frac12)\) \(\approx\) \(2.512465257\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (-1.16 + 1.16i)T - 5iT^{2} \)
7 \( 1 - 3.74T + 7T^{2} \)
11 \( 1 + (-1.64 - 1.64i)T + 11iT^{2} \)
13 \( 1 + (-0.645 + 0.645i)T - 13iT^{2} \)
17 \( 1 + 6.57iT - 17T^{2} \)
19 \( 1 + (-1.41 - 1.41i)T + 19iT^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + (-5.40 - 5.40i)T + 29iT^{2} \)
31 \( 1 + 0.913iT - 31T^{2} \)
37 \( 1 + (1 + i)T + 37iT^{2} \)
41 \( 1 + 6.57T + 41T^{2} \)
43 \( 1 + (-3.74 + 3.74i)T - 43iT^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (7.73 - 7.73i)T - 53iT^{2} \)
59 \( 1 + (9.29 + 9.29i)T + 59iT^{2} \)
61 \( 1 + (10.2 - 10.2i)T - 61iT^{2} \)
67 \( 1 + (-10.8 - 10.8i)T + 67iT^{2} \)
71 \( 1 + 2.70iT - 71T^{2} \)
73 \( 1 + 12.5iT - 73T^{2} \)
79 \( 1 - 14.0iT - 79T^{2} \)
83 \( 1 + (-10.9 + 10.9i)T - 83iT^{2} \)
89 \( 1 - 0.412T + 89T^{2} \)
97 \( 1 - 2.70T + 97T^{2} \)
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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

−8.992331115227246037051488211258, −8.220537996615994210274523589314, −7.44283862673824548234434381740, −6.69318757213752415439060930736, −5.58482129336705470878315318569, −4.91228134632367131012318970314, −4.40445755542395875933978951930, −3.01172859064016368825186161672, −1.89380228257844443205358829284, −1.01706857206708345040071740210, 1.30067307837659604835628955409, 2.08363783911096590762084080864, 3.32301647953333059968779938193, 4.26318022291093245411848435364, 5.12712566368821917191512970208, 6.06175977267467158906727440681, 6.56467720340369688725007410140, 7.75715949889073427754364660573, 8.194476376115388877208887202900, 9.005685766333566072743635585583

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