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Properties

Label 2-48e2-8.5-c1-0-36
Degree $2$
Conductor $2304$
Sign $-0.707 - 0.707i$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·5-s + 6i·13-s − 8·17-s − 11·25-s − 4i·29-s − 2i·37-s − 8·41-s − 7·49-s + 4i·53-s + 10i·61-s + 24·65-s − 6·73-s + 32i·85-s + 16·89-s − 18·97-s + ⋯
L(s)  = 1  − 1.78i·5-s + 1.66i·13-s − 1.94·17-s − 2.20·25-s − 0.742i·29-s − 0.328i·37-s − 1.24·41-s − 49-s + 0.549i·53-s + 1.28i·61-s + 2.97·65-s − 0.702·73-s + 3.47i·85-s + 1.69·89-s − 1.82·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4iT - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 8T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 16T + 89T^{2} \)
97 \( 1 + 18T + 97T^{2} \)
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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

−8.775798061846830041730069472657, −7.959721195406703319186537047535, −6.89633917503043859512751270184, −6.20309480273145978676362427753, −5.13857742396956805203551427282, −4.47108203766416193137878529353, −4.00053368891146151824340777506, −2.26753885309882347217104592797, −1.45365171951982530450713265019, 0, 2.00839927748616696443007480642, 2.94459232243783192743733298219, 3.49871481272708305499429267919, 4.71163983626368025729126972261, 5.72468607478491843059150078010, 6.58400705092942786927186466065, 6.98719002529485757407706228948, 7.896707631749010399782066658556, 8.582556288072288611105926068647

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