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Properties

Label 2-58e2-116.115-c0-0-3
Degree $2$
Conductor $3364$
Sign $-0.483 + 0.875i$
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  − i·2-s − 4-s − 1.24·5-s + i·8-s − 9-s + 1.24i·10-s + 1.80·13-s + 16-s + 0.445i·17-s + i·18-s + 1.24·20-s + 0.554·25-s − 1.80i·26-s − i·32-s + 0.445·34-s + ⋯
L(s)  = 1  − i·2-s − 4-s − 1.24·5-s + i·8-s − 9-s + 1.24i·10-s + 1.80·13-s + 16-s + 0.445i·17-s + i·18-s + 1.24·20-s + 0.554·25-s − 1.80i·26-s − i·32-s + 0.445·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $-0.483 + 0.875i$
Analytic conductor: \(1.67885\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3364} (3363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3364,\ (\ :0),\ -0.483 + 0.875i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7268352985\)
\(L(\frac12)\) \(\approx\) \(0.7268352985\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
29 \( 1 \)
good3 \( 1 + T^{2} \)
5 \( 1 + 1.24T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.80T + T^{2} \)
17 \( 1 - 0.445iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 1.80iT - T^{2} \)
41 \( 1 + 1.80iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 0.445T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 0.445iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 0.445iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.24iT - T^{2} \)
97 \( 1 + 1.80iT - T^{2} \)
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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

−8.820303175819677711108320147582, −8.059531989418342612279016779432, −7.32392923999921096116398685056, −6.03809203046330693206644119561, −5.50515548165918726431697240510, −4.27866351438934572473387657460, −3.75576297328321267267864795537, −3.14392445780036202568085725988, −1.94638853688615600207788197980, −0.56770261362036194640122088488, 1.02534690396700894397675785016, 3.04936706375346806760478416303, 3.68616978478856471736624839313, 4.48874423335333130734082295074, 5.33974764526414701185338951453, 6.22001497856843702437578828506, 6.69922493255771165646505342389, 7.79771334944807594350788863913, 8.160796830061749336195396657960, 8.706429789011640771150054341964

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