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Properties

Label 2-1352-8.3-c0-0-0
Degree $2$
Conductor $1352$
Sign $-i$
Analytic cond. $0.674735$
Root an. cond. $0.821423$
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 − 3-s − 4-s − i·5-s − i·6-s + i·7-s − i·8-s + 10-s + 12-s − 14-s + i·15-s + 16-s + 17-s + i·20-s − i·21-s + ⋯
L(s)  = 1  + i·2-s − 3-s − 4-s − i·5-s − i·6-s + i·7-s − i·8-s + 10-s + 12-s − 14-s + i·15-s + 16-s + 17-s + i·20-s − i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $-i$
Analytic conductor: \(0.674735\)
Root analytic conductor: \(0.821423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6396205520\)
\(L(\frac12)\) \(\approx\) \(0.6396205520\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.806816638732245361690221616289, −8.881672633674710882902928654193, −8.531150172143432585020642384641, −7.53997486762098275141462647353, −6.47226665526910501552028043753, −5.78791058642673620970524756006, −5.17306745944886570076385729980, −4.63410383580908270981363395556, −3.16109920963178993927760226544, −1.13667613107550248963665953153, 0.77865213247262087136783457436, 2.40856117950296044325888258129, 3.48571197112807017857683532284, 4.28621373593704009675708933459, 5.40771038511796594377424597846, 6.11552748492017567784682798082, 7.18721585116344203057469145052, 7.889242502310696734276052574784, 9.079961800622369787575194761504, 10.12523361710602981941221692527

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