Properties

Label 1-1595-1595.1594-r1-0-0
Degree $1$
Conductor $1595$
Sign $1$
Analytic cond. $171.406$
Root an. cond. $171.406$
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 + 12-s + 13-s − 14-s + 16-s − 17-s − 18-s + 19-s + 21-s − 23-s − 24-s − 26-s + 27-s + 28-s − 31-s − 32-s + 34-s + 36-s + 37-s − 38-s + 39-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 12-s + 13-s − 14-s + 16-s − 17-s − 18-s + 19-s + 21-s − 23-s − 24-s − 26-s + 27-s + 28-s − 31-s − 32-s + 34-s + 36-s + 37-s − 38-s + 39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1595 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1595 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1595\)    =    \(5 \cdot 11 \cdot 29\)
Sign: $1$
Analytic conductor: \(171.406\)
Root analytic conductor: \(171.406\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1595} (1594, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1595,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.851061220\)
\(L(\frac12)\) \(\approx\) \(2.851061220\)
\(L(1)\) \(\approx\) \(1.258605170\)
\(L(1)\) \(\approx\) \(1.258605170\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
29 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 - T \)
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

−20.32422738761110806734669080721, −19.64004088711262467258956055577, −18.67308606257744239504878648967, −18.08906117295005369423455951417, −17.709032304251433165554449315541, −16.43964445238081412037201547656, −15.870537167292016032241162754495, −15.154040131675119057188572427196, −14.4084705399220922932724858918, −13.6760249511875549062490229146, −12.762078853615363375901181934954, −11.67696136259021555456648870015, −11.07383083568319737663367949815, −10.2633834218841575034563967289, −9.34478356965383702000704775800, −8.77472357181005596264584211620, −8.03434397515108111932722494276, −7.51942816340483804057279642601, −6.58087361508960945180807414880, −5.56849676743241641329640243474, −4.31501041231563595365235240021, −3.45603867540817036241700955044, −2.37211369468768045764538003587, −1.71193246749524997277366733146, −0.82372923073299959559515880700, 0.82372923073299959559515880700, 1.71193246749524997277366733146, 2.37211369468768045764538003587, 3.45603867540817036241700955044, 4.31501041231563595365235240021, 5.56849676743241641329640243474, 6.58087361508960945180807414880, 7.51942816340483804057279642601, 8.03434397515108111932722494276, 8.77472357181005596264584211620, 9.34478356965383702000704775800, 10.2633834218841575034563967289, 11.07383083568319737663367949815, 11.67696136259021555456648870015, 12.762078853615363375901181934954, 13.6760249511875549062490229146, 14.4084705399220922932724858918, 15.154040131675119057188572427196, 15.870537167292016032241162754495, 16.43964445238081412037201547656, 17.709032304251433165554449315541, 18.08906117295005369423455951417, 18.67308606257744239504878648967, 19.64004088711262467258956055577, 20.32422738761110806734669080721

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