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Properties

Label 1-1157-1157.642-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.182 + 0.983i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.755 + 0.654i)2-s + (−0.800 + 0.599i)3-s + (0.142 + 0.989i)4-s + (−0.841 − 0.540i)5-s + (−0.997 − 0.0713i)6-s + (−0.977 + 0.212i)7-s + (−0.540 + 0.841i)8-s + (0.281 − 0.959i)9-s + (−0.281 − 0.959i)10-s + (−0.540 − 0.841i)11-s + (−0.707 − 0.707i)12-s + (−0.877 − 0.479i)14-s + (0.997 − 0.0713i)15-s + (−0.959 + 0.281i)16-s + (0.755 − 0.654i)17-s + (0.841 − 0.540i)18-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)2-s + (−0.800 + 0.599i)3-s + (0.142 + 0.989i)4-s + (−0.841 − 0.540i)5-s + (−0.997 − 0.0713i)6-s + (−0.977 + 0.212i)7-s + (−0.540 + 0.841i)8-s + (0.281 − 0.959i)9-s + (−0.281 − 0.959i)10-s + (−0.540 − 0.841i)11-s + (−0.707 − 0.707i)12-s + (−0.877 − 0.479i)14-s + (0.997 − 0.0713i)15-s + (−0.959 + 0.281i)16-s + (0.755 − 0.654i)17-s + (0.841 − 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.182 + 0.983i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (642, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.182 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7620380814 + 0.6339221365i\)
\(L(\frac12)\) \(\approx\) \(0.7620380814 + 0.6339221365i\)
\(L(1)\) \(\approx\) \(0.7828119058 + 0.4167674235i\)
\(L(1)\) \(\approx\) \(0.7828119058 + 0.4167674235i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.755 + 0.654i)T \)
3 \( 1 + (-0.800 + 0.599i)T \)
5 \( 1 + (-0.841 - 0.540i)T \)
7 \( 1 + (-0.977 + 0.212i)T \)
11 \( 1 + (-0.540 - 0.841i)T \)
17 \( 1 + (0.755 - 0.654i)T \)
19 \( 1 + (-0.479 - 0.877i)T \)
23 \( 1 + (0.479 + 0.877i)T \)
29 \( 1 + (-0.977 + 0.212i)T \)
31 \( 1 + (0.877 + 0.479i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (0.800 + 0.599i)T \)
43 \( 1 + (0.977 + 0.212i)T \)
47 \( 1 + (-0.142 - 0.989i)T \)
53 \( 1 + (0.989 + 0.142i)T \)
59 \( 1 + (-0.599 + 0.800i)T \)
61 \( 1 + (-0.936 + 0.349i)T \)
67 \( 1 + (0.989 + 0.142i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (0.281 + 0.959i)T \)
79 \( 1 + (-0.281 - 0.959i)T \)
83 \( 1 + (0.997 + 0.0713i)T \)
97 \( 1 + (0.540 - 0.841i)T \)
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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

−21.11047489350289362090219882428, −20.37392612682022878937362307721, −19.340911256562686761844137368566, −18.97056892905266702643656757786, −18.45073949813636376061990952158, −17.248322464412061209621747707671, −16.37444448837170669495222252678, −15.57126992770808181256544892685, −14.86597741857999628286941464666, −13.92878557113476452681244732593, −12.93429258569376999002046183969, −12.44017339219984211491834027975, −11.97805508911711401576293855026, −10.75985027760450643563129181337, −10.525066008674636713412152131591, −9.60037544703053457622823036003, −8.02537955663881960614606598132, −7.20907694527527805148042995905, −6.42674437037664815115871696870, −5.77639865854102409742476776026, −4.62920296749837978421003832146, −3.87974819559154321962383313429, −2.88643995429002596534715368353, −1.94522295094971921237733474266, −0.6060039605127350593724413636, 0.64381991293302651813262696692, 2.894148402690110074705315342233, 3.52981185363591397146008616612, 4.390743693160997319822288326306, 5.3238145580978838771890385664, 5.786049027027338995251250405426, 6.87110734220970549889531890116, 7.561574722955660303626903504944, 8.756355033885988032399208943035, 9.322683967116629118641640606910, 10.62680154212631503956850203858, 11.46044148743652755644570903344, 12.08111033169971728492691214449, 12.86822389828989729796820369559, 13.47509065564439331969265747193, 14.71234852537740007318798671548, 15.636344102501165093221751476897, 15.85398211248939806609634128482, 16.58174837781670050631831009766, 17.13838521721078079202514300092, 18.22311061893130107309439699358, 19.17038382135371575960188147785, 20.05029124954475308586131235572, 21.16680534584200128997335113822, 21.40341235707041485303756225566

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