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Properties

Label 2-58e2-116.115-c0-0-4
Degree $2$
Conductor $3364$
Sign $0.0204 + 0.999i$
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 + 1.80·5-s + i·8-s − 9-s − 1.80i·10-s + 0.445·13-s + 16-s − 1.24i·17-s + i·18-s − 1.80·20-s + 2.24·25-s − 0.445i·26-s − i·32-s − 1.24·34-s + ⋯
L(s)  = 1  − i·2-s − 4-s + 1.80·5-s + i·8-s − 9-s − 1.80i·10-s + 0.445·13-s + 16-s − 1.24i·17-s + i·18-s − 1.80·20-s + 2.24·25-s − 0.445i·26-s − i·32-s − 1.24·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0204 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0204 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $0.0204 + 0.999i$
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: \((2,\ 3364,\ (\ :0),\ 0.0204 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.495877328\)
\(L(\frac12)\) \(\approx\) \(1.495877328\)
\(L(1)\) 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 \( 1 - 1.80T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 0.445T + T^{2} \)
17 \( 1 + 1.24iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 0.445iT - T^{2} \)
41 \( 1 + 0.445iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 1.24T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.24iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.24iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.80iT - T^{2} \)
97 \( 1 + 0.445iT - T^{2} \)
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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

−8.979372071185908769708437823668, −8.231995396246101382083030807335, −7.04056632913461429629816081844, −6.06811454977233388641622922711, −5.47353032645863649510599379627, −4.94984472452217266503797829433, −3.69170921089000496527358048002, −2.66376719528329714745698449667, −2.23826294388706779460837318926, −1.02527779818644100315973285092, 1.33862576861104183472408463601, 2.48730996525649041455365454989, 3.56495948993525169120856894561, 4.69242683717090823710485979733, 5.57616176323697344937081048646, 5.96803710951231364521439274462, 6.45171840347197086021737678172, 7.39692645562042000888120268814, 8.493780451374301941968098231852, 8.753605399941509241436853192165

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