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Properties

Label 1-1113-1113.1112-r0-0-0
Degree $1$
Conductor $1113$
Sign $1$
Analytic cond. $5.16875$
Root an. cond. $5.16875$
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 + 4-s − 5-s + 8-s − 10-s − 11-s − 13-s + 16-s + 17-s + 19-s − 20-s − 22-s + 23-s + 25-s − 26-s − 29-s + 31-s + 32-s + 34-s + 37-s + 38-s − 40-s − 41-s + 43-s − 44-s + 46-s + 47-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s − 11-s − 13-s + 16-s + 17-s + 19-s − 20-s − 22-s + 23-s + 25-s − 26-s − 29-s + 31-s + 32-s + 34-s + 37-s + 38-s − 40-s − 41-s + 43-s − 44-s + 46-s + 47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1113\)    =    \(3 \cdot 7 \cdot 53\)
Sign: $1$
Analytic conductor: \(5.16875\)
Root analytic conductor: \(5.16875\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1113} (1112, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1113,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.454518754\)
\(L(\frac12)\) \(\approx\) \(2.454518754\)
\(L(1)\) \(\approx\) \(1.697217400\)
\(L(1)\) \(\approx\) \(1.697217400\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
53 \( 1 \)
good2 \( 1 + T \)
5 \( 1 - T \)
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 \( 1 + T \)
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

−21.36670488375042174312018077155, −20.65668198482772313490875286783, −20.051339047237063629771962695242, −19.138255607443999735014314432672, −18.61121372210653340696599892283, −17.24207782123772402411821547658, −16.46630477837731987797167239830, −15.76773893590756531990626955695, −15.06978432878000499183898243171, −14.475450580125073204171718432959, −13.47428604181174728319617329523, −12.67874265676334600516518272673, −12.03073848227200757331138851707, −11.34191678457135199790299402303, −10.49457812752734382181873256042, −9.604206027988199876663198134377, −8.1567144932928437121718024676, −7.52555519408557999948268560900, −6.95767113702816188902695371318, −5.56955021887383831307418028464, −5.04700084618276108882116748908, −4.10904441779471928277878605829, −3.146719848706199114793568419119, −2.51899215238905251593398038418, −0.9672415864428210857949871908, 0.9672415864428210857949871908, 2.51899215238905251593398038418, 3.146719848706199114793568419119, 4.10904441779471928277878605829, 5.04700084618276108882116748908, 5.56955021887383831307418028464, 6.95767113702816188902695371318, 7.52555519408557999948268560900, 8.1567144932928437121718024676, 9.604206027988199876663198134377, 10.49457812752734382181873256042, 11.34191678457135199790299402303, 12.03073848227200757331138851707, 12.67874265676334600516518272673, 13.47428604181174728319617329523, 14.475450580125073204171718432959, 15.06978432878000499183898243171, 15.76773893590756531990626955695, 16.46630477837731987797167239830, 17.24207782123772402411821547658, 18.61121372210653340696599892283, 19.138255607443999735014314432672, 20.051339047237063629771962695242, 20.65668198482772313490875286783, 21.36670488375042174312018077155

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