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Properties

Label 2-4368-13.12-c1-0-34
Degree $2$
Conductor $4368$
Sign $0.0805 - 0.996i$
Analytic cond. $34.8786$
Root an. cond. $5.90581$
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  + 3-s + 1.33i·5-s + i·7-s + 9-s + 4.88i·11-s + (3.59 + 0.290i)13-s + 1.33i·15-s + 4.58·17-s + 2.96i·19-s + i·21-s + 6.13·23-s + 3.21·25-s + 27-s − 7.63·29-s − 5.63i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.596i·5-s + 0.377i·7-s + 0.333·9-s + 1.47i·11-s + (0.996 + 0.0805i)13-s + 0.344i·15-s + 1.11·17-s + 0.681i·19-s + 0.218i·21-s + 1.27·23-s + 0.643·25-s + 0.192·27-s − 1.41·29-s − 1.01i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0805 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0805 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4368\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0805 - 0.996i$
Analytic conductor: \(34.8786\)
Root analytic conductor: \(5.90581\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4368} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4368,\ (\ :1/2),\ 0.0805 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.629812652\)
\(L(\frac12)\) \(\approx\) \(2.629812652\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 - iT \)
13 \( 1 + (-3.59 - 0.290i)T \)
good5 \( 1 - 1.33iT - 5T^{2} \)
11 \( 1 - 4.88iT - 11T^{2} \)
17 \( 1 - 4.58T + 17T^{2} \)
19 \( 1 - 2.96iT - 19T^{2} \)
23 \( 1 - 6.13T + 23T^{2} \)
29 \( 1 + 7.63T + 29T^{2} \)
31 \( 1 + 5.63iT - 31T^{2} \)
37 \( 1 - 9.76iT - 37T^{2} \)
41 \( 1 + 3.69iT - 41T^{2} \)
43 \( 1 + 9.63T + 43T^{2} \)
47 \( 1 + 5.27iT - 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 - 5.50iT - 67T^{2} \)
71 \( 1 + 4.88iT - 71T^{2} \)
73 \( 1 + 4.45iT - 73T^{2} \)
79 \( 1 - 3.54T + 79T^{2} \)
83 \( 1 - 5.27iT - 83T^{2} \)
89 \( 1 - 9.94iT - 89T^{2} \)
97 \( 1 - 18.2iT - 97T^{2} \)
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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

−8.474920335851896976926815896582, −7.81980472460530755644575232379, −7.10668692808641321614130907871, −6.54290034193932360407203359000, −5.56409195664351782682356329601, −4.82123374872713086760125247378, −3.76702681592140241257326827665, −3.20976508494331018350502677603, −2.19423054118868240480343631497, −1.36136704438927686188366149170, 0.74593371739718653404676260885, 1.47321558350740286502407246532, 3.02145267087912145335887680169, 3.39498551106206908586479046041, 4.33971355519245775704221425688, 5.32001618355566042637193401150, 5.85616335850839201035565628800, 6.87505953315144945977559932695, 7.54020317824704676734632743557, 8.379553474470218976315483734434

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