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Properties

Label 2-48e2-16.13-c1-0-6
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  + (0.732 − 0.732i)5-s + 2.44i·7-s + (−3.86 + 3.86i)11-s + (3 + 3i)13-s + 3.46·17-s + (−0.378 − 0.378i)19-s − 2.82i·23-s + 3.92i·25-s + (−4.19 − 4.19i)29-s − 7.34·31-s + (1.79 + 1.79i)35-s + (−6.46 + 6.46i)37-s − 11.4i·41-s + (2.44 − 2.44i)43-s − 2.82·47-s + ⋯
L(s)  = 1  + (0.327 − 0.327i)5-s + 0.925i·7-s + (−1.16 + 1.16i)11-s + (0.832 + 0.832i)13-s + 0.840·17-s + (−0.0869 − 0.0869i)19-s − 0.589i·23-s + 0.785i·25-s + (−0.779 − 0.779i)29-s − 1.31·31-s + (0.303 + 0.303i)35-s + (−1.06 + 1.06i)37-s − 1.79i·41-s + (0.373 − 0.373i)43-s − 0.412·47-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

\(L(1)\) \(\approx\) \(1.149163619\)
\(L(\frac12)\) \(\approx\) \(1.149163619\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.732 + 0.732i)T - 5iT^{2} \)
7 \( 1 - 2.44iT - 7T^{2} \)
11 \( 1 + (3.86 - 3.86i)T - 11iT^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + (0.378 + 0.378i)T + 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (4.19 + 4.19i)T + 29iT^{2} \)
31 \( 1 + 7.34T + 31T^{2} \)
37 \( 1 + (6.46 - 6.46i)T - 37iT^{2} \)
41 \( 1 + 11.4iT - 41T^{2} \)
43 \( 1 + (-2.44 + 2.44i)T - 43iT^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + (6.73 - 6.73i)T - 53iT^{2} \)
59 \( 1 + (9.79 - 9.79i)T - 59iT^{2} \)
61 \( 1 + (6.46 + 6.46i)T + 61iT^{2} \)
67 \( 1 + (-0.757 - 0.757i)T + 67iT^{2} \)
71 \( 1 - 16.2iT - 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 2.44T + 79T^{2} \)
83 \( 1 + (1.03 + 1.03i)T + 83iT^{2} \)
89 \( 1 - 8.92iT - 89T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−9.157696812431670537297637579678, −8.705417029956595100995259343878, −7.69015858876111940769240654221, −7.07891122933331850848255209892, −5.93764037741619105062041898338, −5.42495419346922079771826990375, −4.61570600636372675821185646206, −3.53914838496683116228048025740, −2.36285064177276608801447804513, −1.63029110280207490813326234871, 0.37531651768677810398260756793, 1.66998164780330281319300840538, 3.22030272657091140755112100217, 3.44410221861264077576044253812, 4.84699977669482064292151923983, 5.71011642098818158535378422334, 6.21231157576701972513895482179, 7.41984588306915955709582137956, 7.83726151148805465521345517622, 8.629252247215914316419522156236

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