Properties

Label 2-48e2-8.5-c1-0-15
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 2i·13-s + 8i·19-s + 5·25-s + 4·31-s − 10i·37-s − 8i·43-s + 9·49-s − 14i·61-s − 16i·67-s + 10·73-s + 4·79-s + 8i·91-s + 14·97-s + 20·103-s + ⋯
L(s)  = 1  − 1.51·7-s − 0.554i·13-s + 1.83i·19-s + 25-s + 0.718·31-s − 1.64i·37-s − 1.21i·43-s + 1.28·49-s − 1.79i·61-s − 1.95i·67-s + 1.17·73-s + 0.450·79-s + 0.838i·91-s + 1.42·97-s + 1.97·103-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: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.214325323\)
\(L(\frac12)\) \(\approx\) \(1.214325323\)
\(L(\frac{3}{2})\) 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 \( 1 + 4T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 14iT - 61T^{2} \)
67 \( 1 + 16iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−9.008428359699540435016744051413, −8.124169989877371713769418839823, −7.35928195643356245391222192985, −6.44909564911114125700282328747, −5.94757960444598584508574878041, −5.02021312284690983688973493731, −3.74383113409750442137554660928, −3.29021543040717627143857293621, −2.11261587435062822607495516605, −0.54054744680117114954700093718, 0.907548119678773926813708962442, 2.59733218933344662632698734592, 3.14661758904341400413257763534, 4.29872888615737020177104784318, 5.06280644591501156631301521343, 6.26610341501198060980853030440, 6.66926198963039532733097882902, 7.35968717154217309943927251970, 8.554947232483548683002580658808, 9.109355025347877427749698838547

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