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Properties

Label 2-58e2-116.91-c0-0-5
Degree $2$
Conductor $3364$
Sign $0.343 + 0.939i$
Analytic cond. $1.67885$
Root an. cond. $1.29570$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.974 − 0.222i)2-s + (0.900 + 0.433i)4-s + (−0.0990 + 0.433i)5-s + (−0.781 − 0.623i)8-s + (−0.623 + 0.781i)9-s + (0.193 − 0.400i)10-s + (−0.777 − 0.974i)13-s + (0.623 + 0.781i)16-s − 1.80i·17-s + (0.781 − 0.623i)18-s + (−0.277 + 0.347i)20-s + (0.722 + 0.347i)25-s + (0.541 + 1.12i)26-s + (−0.433 − 0.900i)32-s + (−0.400 + 1.75i)34-s + ⋯
L(s)  = 1  + (−0.974 − 0.222i)2-s + (0.900 + 0.433i)4-s + (−0.0990 + 0.433i)5-s + (−0.781 − 0.623i)8-s + (−0.623 + 0.781i)9-s + (0.193 − 0.400i)10-s + (−0.777 − 0.974i)13-s + (0.623 + 0.781i)16-s − 1.80i·17-s + (0.781 − 0.623i)18-s + (−0.277 + 0.347i)20-s + (0.722 + 0.347i)25-s + (0.541 + 1.12i)26-s + (−0.433 − 0.900i)32-s + (−0.400 + 1.75i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $0.343 + 0.939i$
Analytic conductor: \(1.67885\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3364} (2759, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3364,\ (\ :0),\ 0.343 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5840750918\)
\(L(\frac12)\) \(\approx\) \(0.5840750918\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.974 + 0.222i)T \)
29 \( 1 \)
good3 \( 1 + (0.623 - 0.781i)T^{2} \)
5 \( 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)T^{2} \)
7 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)T^{2} \)
17 \( 1 + 1.80iT - T^{2} \)
19 \( 1 + (0.623 + 0.781i)T^{2} \)
23 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (-0.900 - 0.433i)T^{2} \)
37 \( 1 + (0.974 + 0.777i)T + (0.222 + 0.974i)T^{2} \)
41 \( 1 + 1.24iT - T^{2} \)
43 \( 1 + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.222 + 0.974i)T^{2} \)
53 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.781 - 1.62i)T + (-0.623 + 0.781i)T^{2} \)
67 \( 1 + (0.222 + 0.974i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (-1.75 + 0.400i)T + (0.900 - 0.433i)T^{2} \)
79 \( 1 + (-0.222 - 0.974i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.433 - 0.0990i)T + (0.900 + 0.433i)T^{2} \)
97 \( 1 + (0.541 - 1.12i)T + (-0.623 - 0.781i)T^{2} \)
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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

−8.682487375725434083964644591817, −7.939178991286299186090407198080, −7.22069911868347116960200733966, −6.87664516708936405667236485105, −5.53114744604239880219392343469, −5.10486405529229733403104855899, −3.60938549436913773905229196575, −2.76291917841630634406854080912, −2.21087949270746761035978958506, −0.51653215604769184355076783554, 1.17016847064011474559253113649, 2.21137288541667874376231964288, 3.28398883045224369517308600235, 4.34297543075486433320507045086, 5.32237871382586041824772167431, 6.27858093730068970362264529830, 6.62701414249471753688980251724, 7.61704356875073313757185443011, 8.398114391014519469470200275753, 8.829734248091205946310940416889

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