↑

Properties

Label 1-1309-1309.1172-r0-0-0
Degree $1$
Conductor $1309$
Sign $0.701 - 0.712i$
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  + (0.104 + 0.994i)2-s + (0.669 − 0.743i)3-s + (−0.978 + 0.207i)4-s + (0.913 − 0.406i)5-s + (0.809 + 0.587i)6-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.5 + 0.866i)10-s + (−0.5 + 0.866i)12-s + (−0.809 + 0.587i)13-s + (0.309 − 0.951i)15-s + (0.913 − 0.406i)16-s + (0.978 − 0.207i)18-s + (−0.978 − 0.207i)19-s + (−0.809 + 0.587i)20-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (0.669 − 0.743i)3-s + (−0.978 + 0.207i)4-s + (0.913 − 0.406i)5-s + (0.809 + 0.587i)6-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.5 + 0.866i)10-s + (−0.5 + 0.866i)12-s + (−0.809 + 0.587i)13-s + (0.309 − 0.951i)15-s + (0.913 − 0.406i)16-s + (0.978 − 0.207i)18-s + (−0.978 − 0.207i)19-s + (−0.809 + 0.587i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1309\)    =    \(7 \cdot 11 \cdot 17\)
Sign: $0.701 - 0.712i$
Analytic conductor: \(6.07897\)
Root analytic conductor: \(6.07897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1309} (1172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1309,\ (0:\ ),\ 0.701 - 0.712i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.683665614 - 0.7056403852i\)
\(L(\frac12)\) \(\approx\) \(1.683665614 - 0.7056403852i\)
\(L(1)\) \(\approx\) \(1.332889692 + 0.02275253895i\)
\(L(1)\) \(\approx\) \(1.332889692 + 0.02275253895i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.104 + 0.994i)T \)
3 \( 1 + (0.669 - 0.743i)T \)
5 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (-0.669 - 0.743i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (0.978 + 0.207i)T \)
53 \( 1 + (0.913 + 0.406i)T \)
59 \( 1 + (0.978 - 0.207i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.978 - 0.207i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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

−21.26659360065618455726373900993, −20.34616723235245770598333138778, −19.69166890856575251369723894000, −19.00241951846862421119157776712, −18.19433059979867649028692954374, −17.27260456597806637389203438086, −16.7797897296075555767647759683, −15.26261933843542527057480446100, −14.886657275958421799271804168977, −14.019034350022787366964395884, −13.42214621607570695126825986823, −12.70579527200832319131313076082, −11.62790384608335585844492129426, −10.66372781225015110126899878147, −10.17820380154241671534398459355, −9.594646237438435086111652807, −8.77396336057933197210239174712, −7.996682667295795456935409677469, −6.72373118496726764772057921714, −5.44014818994288590498677282464, −4.94388111511103933210029021038, −3.8610632296391723678058038607, −2.967289789929730307214130202967, −2.40171487427944831688368096230, −1.42141988194974152672956859603, 0.62068151646151781415531532664, 1.925862546269468590609942721978, 2.758382936578562508215116606536, 4.09626035510638529244735592827, 4.89690181764203197561085949462, 5.93142148389896541608925504516, 6.65136834186134195047770781561, 7.240358000786545094381684577480, 8.372225471823826550717844637457, 8.81583762710262765657381145392, 9.59036272259536712032146439504, 10.4222691670346575302483158662, 12.1099785125581535328495246404, 12.5521260116024361106585025960, 13.494168063033403706776915865, 13.88385843150326783522033458980, 14.68378025463253308008676804757, 15.30475818324306570068236955840, 16.41081652010095560724465069543, 17.27911147634331075520025161944, 17.50257158749700183968918183359, 18.60191860365280884925045995499, 19.08799741199378015439795108425, 20.009076322776392015267358863872, 21.133194016644131952047570116662

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