Properties

Label 2-3332-17.16-c1-0-5
Degree $2$
Conductor $3332$
Sign $-0.840 - 0.542i$
Analytic cond. $26.6061$
Root an. cond. $5.15811$
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  − 0.381i·3-s + 2.61i·5-s + 2.85·9-s − 3.46i·11-s − 2.14·13-s + 15-s + (−3.46 − 2.23i)17-s − 5.60·19-s + 5.60i·23-s − 1.85·25-s − 2.23i·27-s + 7.74i·29-s + 1.85i·31-s − 1.32·33-s + 1.32i·37-s + ⋯
L(s)  = 1  − 0.220i·3-s + 1.17i·5-s + 0.951·9-s − 1.04i·11-s − 0.593·13-s + 0.258·15-s + (−0.840 − 0.542i)17-s − 1.28·19-s + 1.16i·23-s − 0.370·25-s − 0.430i·27-s + 1.43i·29-s + 0.333i·31-s − 0.230·33-s + 0.217i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.840 - 0.542i$
Analytic conductor: \(26.6061\)
Root analytic conductor: \(5.15811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :1/2),\ -0.840 - 0.542i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7613852725\)
\(L(\frac12)\) \(\approx\) \(0.7613852725\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
17 \( 1 + (3.46 + 2.23i)T \)
good3 \( 1 + 0.381iT - 3T^{2} \)
5 \( 1 - 2.61iT - 5T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + 2.14T + 13T^{2} \)
19 \( 1 + 5.60T + 19T^{2} \)
23 \( 1 - 5.60iT - 23T^{2} \)
29 \( 1 - 7.74iT - 29T^{2} \)
31 \( 1 - 1.85iT - 31T^{2} \)
37 \( 1 - 1.32iT - 37T^{2} \)
41 \( 1 - 0.381iT - 41T^{2} \)
43 \( 1 + 0.854T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + 1.85T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 8.56iT - 61T^{2} \)
67 \( 1 + 7.56T + 67T^{2} \)
71 \( 1 + 1.32iT - 71T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 - 7.74iT - 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 7.85iT - 97T^{2} \)
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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

−8.924512361925272452571974562552, −8.111269664040519195436702448473, −7.18557227142769825872816760763, −6.85965335443563396925250393904, −6.13915369708728576619654714346, −5.13972483001251989712241037319, −4.21666664647768923701789472020, −3.29822668015596840147180587136, −2.55088805469103063641341980893, −1.45399751469632690037250208671, 0.21991285521800350357138704208, 1.66745383720972612140427375616, 2.36802483371766203422864958819, 3.99321275277394649916316388102, 4.55620175884211994595985532122, 4.86817197436655579143017265528, 6.16101778698245373208729904671, 6.76956551243054078412338871980, 7.71114935806115214849002836008, 8.339189131075492524827099326526

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