↑

Properties

Label 1-1309-1309.1081-r0-0-0
Degree $1$
Conductor $1309$
Sign $-0.747 - 0.664i$
Analytic cond. $6.07897$
Root an. cond. $6.07897$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0523 − 0.998i)2-s + (0.0261 − 0.999i)3-s + (−0.994 − 0.104i)4-s + (0.983 − 0.182i)5-s + (−0.996 − 0.0784i)6-s + (−0.156 + 0.987i)8-s + (−0.998 − 0.0523i)9-s + (−0.130 − 0.991i)10-s + (−0.130 + 0.991i)12-s + (0.951 + 0.309i)13-s + (−0.156 − 0.987i)15-s + (0.978 + 0.207i)16-s + (−0.104 + 0.994i)18-s + (0.629 − 0.777i)19-s + (−0.996 + 0.0784i)20-s + ⋯
L(s)  = 1  + (0.0523 − 0.998i)2-s + (0.0261 − 0.999i)3-s + (−0.994 − 0.104i)4-s + (0.983 − 0.182i)5-s + (−0.996 − 0.0784i)6-s + (−0.156 + 0.987i)8-s + (−0.998 − 0.0523i)9-s + (−0.130 − 0.991i)10-s + (−0.130 + 0.991i)12-s + (0.951 + 0.309i)13-s + (−0.156 − 0.987i)15-s + (0.978 + 0.207i)16-s + (−0.104 + 0.994i)18-s + (0.629 − 0.777i)19-s + (−0.996 + 0.0784i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1309\)    =    \(7 \cdot 11 \cdot 17\)
Sign: $-0.747 - 0.664i$
Analytic conductor: \(6.07897\)
Root analytic conductor: \(6.07897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1309} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1309,\ (0:\ ),\ -0.747 - 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6557836657 - 1.725426157i\)
\(L(\frac12)\) \(\approx\) \(0.6557836657 - 1.725426157i\)
\(L(1)\) \(\approx\) \(0.8498549514 - 0.9263698247i\)
\(L(1)\) \(\approx\) \(0.8498549514 - 0.9263698247i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.0523 - 0.998i)T \)
3 \( 1 + (0.0261 - 0.999i)T \)
5 \( 1 + (0.983 - 0.182i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.629 - 0.777i)T \)
23 \( 1 + (0.608 + 0.793i)T \)
29 \( 1 + (0.972 + 0.233i)T \)
31 \( 1 + (-0.566 - 0.824i)T \)
37 \( 1 + (0.725 + 0.688i)T \)
41 \( 1 + (0.972 - 0.233i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (0.994 - 0.104i)T \)
53 \( 1 + (0.838 + 0.544i)T \)
59 \( 1 + (-0.629 - 0.777i)T \)
61 \( 1 + (-0.824 - 0.566i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.760 + 0.649i)T \)
73 \( 1 + (-0.284 - 0.958i)T \)
79 \( 1 + (-0.942 + 0.333i)T \)
83 \( 1 + (0.453 - 0.891i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.0784 - 0.996i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.387903513281195334635722177115, −20.8011734479758059516280450811, −19.8436495240708699427793181321, −18.59910274475243275932907866622, −18.04553563936959002163982271855, −17.298644088934244790537644016807, −16.50073964506782120567101226582, −16.04976938506615945430798806700, −15.13738313800335739735385088496, −14.442232124718327602124307045749, −13.87841690712142659343855235203, −13.11323357633642181653236154354, −12.08864813915754063048051886412, −10.709906001323227559583559870578, −10.3123925366119643192479647699, −9.32188174200231064342655192264, −8.83407781599599049458498847639, −7.95451145030255486656908478229, −6.79219629697727111164287360389, −5.93171430647290635724171044807, −5.45668628855221714504599069945, −4.52788627690890222234848458624, −3.6094069289394892496032770140, −2.70693217983830629919502268696, −1.07976530785841794356513780482, 0.9310343752107924318116013274, 1.53126932643832404539027624027, 2.52145206463760091140031873516, 3.21144879000244328446386720759, 4.51903603564758375657716319926, 5.51763468814271983382175166054, 6.145373840283429680351816161708, 7.20050447390811448991387837223, 8.2859651371568888138270876630, 9.05251178323054381350120641371, 9.62284092442760967238206207768, 10.78604875579872243054734108161, 11.37484971881939490120550064884, 12.21045683430523673764001263603, 13.093814692301890425030516932715, 13.50338626228915319084106388470, 14.04732163697736828582450421773, 14.99308761010081234778932403934, 16.35832735766123503821011035763, 17.321644822274750833877419898, 17.78847505709906700507870210464, 18.50005627072597566984736098652, 19.04772331562224497528601213629, 20.08994322190308135033013846445, 20.44362096117256943431074329009

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