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Properties

Label 2-4368-13.12-c1-0-49
Degree $2$
Conductor $4368$
Sign $0.954 - 0.298i$
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 + 2.79i·5-s + i·7-s + 9-s − 1.93i·11-s + (−1.07 − 3.44i)13-s + 2.79i·15-s + 4.51·17-s − 6.15i·19-s + i·21-s + 1.79·23-s − 2.79·25-s + 27-s + 8.67·29-s + 4.87i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.24i·5-s + 0.377i·7-s + 0.333·9-s − 0.582i·11-s + (−0.298 − 0.954i)13-s + 0.720i·15-s + 1.09·17-s − 1.41i·19-s + 0.218i·21-s + 0.374·23-s − 0.558·25-s + 0.192·27-s + 1.61·29-s + 0.875i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4368\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.954 - 0.298i$
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: \((2,\ 4368,\ (\ :1/2),\ 0.954 - 0.298i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.568070151\)
\(L(\frac12)\) \(\approx\) \(2.568070151\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.07 + 3.44i)T \)
good5 \( 1 - 2.79iT - 5T^{2} \)
11 \( 1 + 1.93iT - 11T^{2} \)
17 \( 1 - 4.51T + 17T^{2} \)
19 \( 1 + 6.15iT - 19T^{2} \)
23 \( 1 - 1.79T + 23T^{2} \)
29 \( 1 - 8.67T + 29T^{2} \)
31 \( 1 - 4.87iT - 31T^{2} \)
37 \( 1 - 9.96iT - 37T^{2} \)
41 \( 1 + 5.88iT - 41T^{2} \)
43 \( 1 - 3.42T + 43T^{2} \)
47 \( 1 + 12.1iT - 47T^{2} \)
53 \( 1 + 0.361T + 53T^{2} \)
59 \( 1 - 5.51iT - 59T^{2} \)
61 \( 1 + 2.15T + 61T^{2} \)
67 \( 1 + 9.24iT - 67T^{2} \)
71 \( 1 + 6.06iT - 71T^{2} \)
73 \( 1 + 2.86iT - 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + 3.67iT - 83T^{2} \)
89 \( 1 + 3.44iT - 89T^{2} \)
97 \( 1 - 14.6iT - 97T^{2} \)
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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

−8.370098209726659546673424296572, −7.69684441843758729368264184327, −6.92369673491746379116331710585, −6.43162444491638914546879801017, −5.42486524952214215719066003508, −4.74187256424366911107249944904, −3.31496649429698423777134171365, −3.11426770608689506766462843461, −2.34372345705825321879420004277, −0.855130711577169740974292419302, 0.977788208870383573784361447104, 1.73927077859868322350411666010, 2.84538246886871332451251011633, 4.01791913162456225251526610303, 4.40402850668747403293117587957, 5.25969591478137857422486059996, 6.09573720478159381136402474904, 7.04127771032387208560297887467, 7.82590910350135596523500525588, 8.218400627027326970022242213681

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