Properties

Label 2-4608-8.5-c1-0-26
Degree $2$
Conductor $4608$
Sign $-i$
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  + 1.41i·5-s + 3.16·7-s + 4.47i·11-s − 4.47i·13-s − 6.32·17-s − 2.82i·19-s − 4·23-s + 2.99·25-s + 4.24i·29-s + 3.16·31-s + 4.47i·35-s + 4.47i·37-s + 6.32·41-s + 8.48i·43-s + 12·47-s + ⋯
L(s)  = 1  + 0.632i·5-s + 1.19·7-s + 1.34i·11-s − 1.24i·13-s − 1.53·17-s − 0.648i·19-s − 0.834·23-s + 0.599·25-s + 0.787i·29-s + 0.567·31-s + 0.755i·35-s + 0.735i·37-s + 0.987·41-s + 1.29i·43-s + 1.75·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $-i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.831507606\)
\(L(\frac12)\) \(\approx\) \(1.831507606\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.41iT - 5T^{2} \)
7 \( 1 - 3.16T + 7T^{2} \)
11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 + 4.47iT - 13T^{2} \)
17 \( 1 + 6.32T + 17T^{2} \)
19 \( 1 + 2.82iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 4.24iT - 29T^{2} \)
31 \( 1 - 3.16T + 31T^{2} \)
37 \( 1 - 4.47iT - 37T^{2} \)
41 \( 1 - 6.32T + 41T^{2} \)
43 \( 1 - 8.48iT - 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 - 7.07iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 13.4iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + 3.16T + 79T^{2} \)
83 \( 1 - 4.47iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.443812398675053348664916355818, −7.66377600469944258895058734605, −7.18255118204579517282291752951, −6.42092672618723071637751776786, −5.50770686199471315678873069297, −4.61664237389383047779993652450, −4.29456653321917394472522471413, −2.86188023339142060324960684881, −2.31083669425428653835392471330, −1.19191759317011121364563067253, 0.52506369006935300273114816764, 1.70638651099839021171217363175, 2.41689577253781448234172571022, 3.86737739404645600984514664345, 4.34037098460167665946891508729, 5.12725931762562092761061329704, 5.93775521018433520203131553285, 6.59572343860576262282339869401, 7.57112539864984449845382902628, 8.278275737267228641379213820684

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