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Properties

Label 2-48e2-16.13-c1-0-8
Degree $2$
Conductor $2304$
Sign $-0.608 - 0.793i$
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  + 5.27i·7-s + (4.46 + 4.46i)13-s + (−2.44 − 2.44i)19-s + 5i·25-s − 4.52·31-s + (8.46 − 8.46i)37-s + (−7.34 + 7.34i)43-s − 20.8·49-s + (−3.53 − 3.53i)61-s + (11.3 + 11.3i)67-s − 13.8i·73-s − 15.0·79-s + (−23.5 + 23.5i)91-s − 13.8·97-s + 16.5i·103-s + ⋯
L(s)  = 1  + 1.99i·7-s + (1.23 + 1.23i)13-s + (−0.561 − 0.561i)19-s + i·25-s − 0.811·31-s + (1.39 − 1.39i)37-s + (−1.12 + 1.12i)43-s − 2.97·49-s + (−0.452 − 0.452i)61-s + (1.38 + 1.38i)67-s − 1.62i·73-s − 1.69·79-s + (−2.46 + 2.46i)91-s − 1.40·97-s + 1.63i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ -0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.465710233\)
\(L(\frac12)\) \(\approx\) \(1.465710233\)
\(L(\frac{3}{2})\) 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 \( 1 - 5iT^{2} \)
7 \( 1 - 5.27iT - 7T^{2} \)
11 \( 1 - 11iT^{2} \)
13 \( 1 + (-4.46 - 4.46i)T + 13iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (2.44 + 2.44i)T + 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29iT^{2} \)
31 \( 1 + 4.52T + 31T^{2} \)
37 \( 1 + (-8.46 + 8.46i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (7.34 - 7.34i)T - 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 + (3.53 + 3.53i)T + 61iT^{2} \)
67 \( 1 + (-11.3 - 11.3i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 13.8iT - 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−9.066393504993887707494606184668, −8.777757881685222677921527887157, −7.905909933915035093510268933515, −6.77164150660328614503371880843, −6.10036591817995130557015722415, −5.48721553972871912153376723933, −4.53314052634464272074898086895, −3.48698054325897862424510666638, −2.44610968035787209939317576470, −1.63357317184786226376672506814, 0.51284004097641588083151319081, 1.49240085889563123572732993797, 3.10562037065779567244079280093, 3.86453175755610812304201706368, 4.51081874264219093132046641993, 5.66400124466060795312092612287, 6.49412983939855436233226378033, 7.17651997064807759198413966953, 8.123219976621195283483764478745, 8.368034386339189546653311005393

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