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Properties

Label 2-1339-1339.1338-c0-0-5
Degree $2$
Conductor $1339$
Sign $1$
Analytic cond. $0.668248$
Root an. cond. $0.817464$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 1.41·5-s + 9-s − 1.41·11-s − 13-s + 16-s + 1.41·20-s + 1.00·25-s − 2·29-s − 1.41·31-s + 36-s − 1.41·37-s − 1.41·44-s + 1.41·45-s − 1.41·47-s + 49-s − 52-s − 2.00·55-s + 64-s − 1.41·65-s + 1.41·67-s + 1.41·71-s + 1.41·73-s + 1.41·80-s + 81-s + 1.41·89-s − 1.41·99-s + ⋯
L(s)  = 1  + 4-s + 1.41·5-s + 9-s − 1.41·11-s − 13-s + 16-s + 1.41·20-s + 1.00·25-s − 2·29-s − 1.41·31-s + 36-s − 1.41·37-s − 1.41·44-s + 1.41·45-s − 1.41·47-s + 49-s − 52-s − 2.00·55-s + 64-s − 1.41·65-s + 1.41·67-s + 1.41·71-s + 1.41·73-s + 1.41·80-s + 81-s + 1.41·89-s − 1.41·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1339\)    =    \(13 \cdot 103\)
Sign: $1$
Analytic conductor: \(0.668248\)
Root analytic conductor: \(0.817464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1339} (1338, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1339,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.600967572\)
\(L(\frac12)\) \(\approx\) \(1.600967572\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + T \)
103 \( 1 + T \)
good2 \( 1 - T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 - 1.41T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 1.41T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + 1.41T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 - 1.41T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
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

−9.860435648138430227816745354748, −9.343978265827071423791042694218, −7.955429592021865594218418966463, −7.27410119817156664797620640716, −6.63611406242812736517912606607, −5.48366736896756247957812664238, −5.19491827795252023426466386141, −3.55074172187910242443305736986, −2.27719345767316970553303491386, −1.85419485339687950988667764041, 1.85419485339687950988667764041, 2.27719345767316970553303491386, 3.55074172187910242443305736986, 5.19491827795252023426466386141, 5.48366736896756247957812664238, 6.63611406242812736517912606607, 7.27410119817156664797620640716, 7.955429592021865594218418966463, 9.343978265827071423791042694218, 9.860435648138430227816745354748

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