Properties

Label 2-4608-24.11-c1-0-21
Degree $2$
Conductor $4608$
Sign $-0.816 - 0.577i$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
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  + 4.24i·7-s + 4i·11-s + 6i·13-s + 4.24i·17-s + 2.82·19-s + 6·23-s − 5·25-s + 8.48·29-s + 4.24i·31-s − 6i·37-s + 1.41i·41-s − 2.82·43-s + 6·47-s − 10.9·49-s − 8.48·53-s + ⋯
L(s)  = 1  + 1.60i·7-s + 1.20i·11-s + 1.66i·13-s + 1.02i·17-s + 0.648·19-s + 1.25·23-s − 25-s + 1.57·29-s + 0.762i·31-s − 0.986i·37-s + 0.220i·41-s − 0.431·43-s + 0.875·47-s − 1.57·49-s − 1.16·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $-0.816 - 0.577i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (2303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ -0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.807130667\)
\(L(\frac12)\) \(\approx\) \(1.807130667\)
\(L(\frac{3}{2})\) 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 \( 1 + 5T^{2} \)
7 \( 1 - 4.24iT - 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 - 4.24iT - 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + 2.82T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 8.48T + 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 + 6iT - 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 4.24iT - 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 - 12.7iT - 89T^{2} \)
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

−8.709171656332828031662951593111, −7.991353971125492109704267672222, −7.00370890513115341823744323167, −6.50831106360807919756623427913, −5.67411040927086991868436406844, −4.90144293572366410068821107480, −4.25940407479232988917759527689, −3.14405262904506503251786177068, −2.18563953736504578282034994336, −1.59980489938982102839267121338, 0.58395077923372593948786094163, 1.04728281394661421644117801499, 2.89201637154261475087014709040, 3.25878743773318187017375951468, 4.25083217142396390543041661476, 5.08522473669796255826782494454, 5.78491311749847362212961810698, 6.68291336864496306588983911050, 7.39616480884917692908289590254, 7.936490798915615955375405448723

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