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Properties

Label 1-5345-5345.5344-r0-0-0
Degree $1$
Conductor $5345$
Sign $1$
Analytic cond. $24.8220$
Root an. cond. $24.8220$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 11-s − 12-s − 13-s + 14-s + 16-s − 17-s + 18-s + 19-s − 21-s − 22-s + 23-s − 24-s − 26-s − 27-s + 28-s + 29-s − 31-s + 32-s + 33-s − 34-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 11-s − 12-s − 13-s + 14-s + 16-s − 17-s + 18-s + 19-s − 21-s − 22-s + 23-s − 24-s − 26-s − 27-s + 28-s + 29-s − 31-s + 32-s + 33-s − 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5345 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5345 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5345\)    =    \(5 \cdot 1069\)
Sign: $1$
Analytic conductor: \(24.8220\)
Root analytic conductor: \(24.8220\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5345} (5344, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 5345,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.778104413\)
\(L(\frac12)\) \(\approx\) \(2.778104413\)
\(L(1)\) \(\approx\) \(1.597644334\)
\(L(1)\) \(\approx\) \(1.597644334\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
1069 \( 1 \)
good2 \( 1 + T \)
3 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.74227183490522477622454822616, −17.26894950128618156727342321798, −16.544839805068017156873063867181, −15.758570843493242186479824210913, −15.36339093504391374287604718243, −14.67363521878867992646245673488, −13.79692579942690112222212557420, −13.3364577703764547149428582187, −12.32004373297802307416134524203, −12.20339861164152203816132461381, −11.20979977668504263153012951392, −10.863120619182009063873966485892, −10.27392318658248030381744891704, −9.306284494659195079287230549701, −8.18399571995203346553285871815, −7.33485331702665164229540652031, −7.0902149874791224058031632822, −6.10439819399898621471849842475, −5.18155455221520311036282621789, −5.05961208193672495077311587096, −4.42636406928845201917009458524, −3.39724838286083830276778065776, −2.43436070510551147703764917818, −1.79370687041776143363263847876, −0.76298606138668372188654188941, 0.76298606138668372188654188941, 1.79370687041776143363263847876, 2.43436070510551147703764917818, 3.39724838286083830276778065776, 4.42636406928845201917009458524, 5.05961208193672495077311587096, 5.18155455221520311036282621789, 6.10439819399898621471849842475, 7.0902149874791224058031632822, 7.33485331702665164229540652031, 8.18399571995203346553285871815, 9.306284494659195079287230549701, 10.27392318658248030381744891704, 10.863120619182009063873966485892, 11.20979977668504263153012951392, 12.20339861164152203816132461381, 12.32004373297802307416134524203, 13.3364577703764547149428582187, 13.79692579942690112222212557420, 14.67363521878867992646245673488, 15.36339093504391374287604718243, 15.758570843493242186479824210913, 16.544839805068017156873063867181, 17.26894950128618156727342321798, 17.74227183490522477622454822616

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