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Properties

Label 1-1596-1596.1595-r0-0-0
Degree $1$
Conductor $1596$
Sign $1$
Analytic cond. $7.41179$
Root an. cond. $7.41179$
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  + 5-s + 11-s + 13-s + 17-s + 23-s + 25-s + 29-s − 31-s − 37-s − 41-s − 43-s − 47-s + 53-s + 55-s + 59-s − 61-s + 65-s + 67-s − 71-s − 73-s + 79-s − 83-s + 85-s − 89-s + 97-s + ⋯
L(s)  = 1  + 5-s + 11-s + 13-s + 17-s + 23-s + 25-s + 29-s − 31-s − 37-s − 41-s − 43-s − 47-s + 53-s + 55-s + 59-s − 61-s + 65-s + 67-s − 71-s − 73-s + 79-s − 83-s + 85-s − 89-s + 97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1596\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(7.41179\)
Root analytic conductor: \(7.41179\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1596} (1595, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1596,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.357249200\)
\(L(\frac12)\) \(\approx\) \(2.357249200\)
\(L(1)\) \(\approx\) \(1.477149138\)
\(L(1)\) \(\approx\) \(1.477149138\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good5 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 + T \)
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

−20.61051033173300938105428073615, −19.76457433093281947093049219268, −18.87957380735386545845681257524, −18.25697051325790834584270545548, −17.46379533169645665222838837204, −16.792571383885244626298140869874, −16.21026852491124520528491550421, −15.075655831517118237088113462827, −14.42364240667533023857860834240, −13.71974655626475725754569346536, −13.071257507962280412119921299897, −12.17301251482545582384406481147, −11.37530068555345005681746489677, −10.4676956170528156889242982272, −9.82438879412605716576608540997, −8.93457788228859693118226406654, −8.42227941441989171203784186689, −7.089409436706318144664458849845, −6.49768270127274173007907432755, −5.65452147976532109343242170488, −4.927919921458490531436395540101, −3.7185011865359851155942780990, −3.01766342012670891018528739480, −1.722611706042244911789822163086, −1.13987390994379447923369579345, 1.13987390994379447923369579345, 1.722611706042244911789822163086, 3.01766342012670891018528739480, 3.7185011865359851155942780990, 4.927919921458490531436395540101, 5.65452147976532109343242170488, 6.49768270127274173007907432755, 7.089409436706318144664458849845, 8.42227941441989171203784186689, 8.93457788228859693118226406654, 9.82438879412605716576608540997, 10.4676956170528156889242982272, 11.37530068555345005681746489677, 12.17301251482545582384406481147, 13.071257507962280412119921299897, 13.71974655626475725754569346536, 14.42364240667533023857860834240, 15.075655831517118237088113462827, 16.21026852491124520528491550421, 16.792571383885244626298140869874, 17.46379533169645665222838837204, 18.25697051325790834584270545548, 18.87957380735386545845681257524, 19.76457433093281947093049219268, 20.61051033173300938105428073615

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