Properties

Label 2-48e2-16.13-c1-0-30
Degree $2$
Conductor $2304$
Sign $-0.382 + 0.923i$
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  + 1.41i·7-s + (1 + i)13-s − 6·17-s + (−4.24 − 4.24i)19-s − 8.48i·23-s + 5i·25-s + (−6 − 6i)29-s + 1.41·31-s + (5 − 5i)37-s + 6i·41-s + (4.24 − 4.24i)43-s − 8.48·47-s + 5·49-s + (6 − 6i)53-s + (5 + 5i)61-s + ⋯
L(s)  = 1  + 0.534i·7-s + (0.277 + 0.277i)13-s − 1.45·17-s + (−0.973 − 0.973i)19-s − 1.76i·23-s + i·25-s + (−1.11 − 1.11i)29-s + 0.254·31-s + (0.821 − 0.821i)37-s + 0.937i·41-s + (0.646 − 0.646i)43-s − 1.23·47-s + 0.714·49-s + (0.824 − 0.824i)53-s + (0.640 + 0.640i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.382 + 0.923i$
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: \((2,\ 2304,\ (\ :1/2),\ -0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8244834906\)
\(L(\frac12)\) \(\approx\) \(0.8244834906\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.41iT - 7T^{2} \)
11 \( 1 - 11iT^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + (4.24 + 4.24i)T + 19iT^{2} \)
23 \( 1 + 8.48iT - 23T^{2} \)
29 \( 1 + (6 + 6i)T + 29iT^{2} \)
31 \( 1 - 1.41T + 31T^{2} \)
37 \( 1 + (-5 + 5i)T - 37iT^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 + (-4.24 + 4.24i)T - 43iT^{2} \)
47 \( 1 + 8.48T + 47T^{2} \)
53 \( 1 + (-6 + 6i)T - 53iT^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 + (-5 - 5i)T + 61iT^{2} \)
67 \( 1 + (2.82 + 2.82i)T + 67iT^{2} \)
71 \( 1 + 8.48iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 1.41T + 79T^{2} \)
83 \( 1 + (8.48 + 8.48i)T + 83iT^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−8.845914283096735284626699754425, −8.163451320712628939103101405889, −7.07897673077629750437062174058, −6.46714280787397210439488723293, −5.71073416294572733335386709075, −4.61527511519245317811381811738, −4.07282440069341845717894324556, −2.67615584958346066760209047680, −2.03104027374006195044833112275, −0.27587155425655764035961868494, 1.35511617079620690035520217638, 2.45647611717807422851071426984, 3.70027810948321612485881886747, 4.27062819085745809150463364375, 5.34444544885517623226941439621, 6.17597995783632056657631792267, 6.92571965918598632690027979412, 7.72794468623125258241545305254, 8.465640868610099452471933417458, 9.203497233206157049903494076163

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