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Properties

Label 1-1157-1157.1023-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.803 + 0.595i$
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  + (0.0475 − 0.998i)2-s + (0.995 − 0.0950i)3-s + (−0.995 − 0.0950i)4-s + (−0.142 + 0.989i)5-s + (−0.0475 − 0.998i)6-s + (0.786 + 0.618i)7-s + (−0.142 + 0.989i)8-s + (0.981 − 0.189i)9-s + (0.981 + 0.189i)10-s + (−0.786 + 0.618i)11-s − 12-s + (0.654 − 0.755i)14-s + (−0.0475 + 0.998i)15-s + (0.981 + 0.189i)16-s + (0.0475 + 0.998i)17-s + (−0.142 − 0.989i)18-s + ⋯
L(s)  = 1  + (0.0475 − 0.998i)2-s + (0.995 − 0.0950i)3-s + (−0.995 − 0.0950i)4-s + (−0.142 + 0.989i)5-s + (−0.0475 − 0.998i)6-s + (0.786 + 0.618i)7-s + (−0.142 + 0.989i)8-s + (0.981 − 0.189i)9-s + (0.981 + 0.189i)10-s + (−0.786 + 0.618i)11-s − 12-s + (0.654 − 0.755i)14-s + (−0.0475 + 0.998i)15-s + (0.981 + 0.189i)16-s + (0.0475 + 0.998i)17-s + (−0.142 − 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.803 + 0.595i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1023, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.803 + 0.595i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.678870745 + 0.5548914992i\)
\(L(\frac12)\) \(\approx\) \(1.678870745 + 0.5548914992i\)
\(L(1)\) \(\approx\) \(1.313965548 - 0.1325846733i\)
\(L(1)\) \(\approx\) \(1.313965548 - 0.1325846733i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.0475 - 0.998i)T \)
3 \( 1 + (0.995 - 0.0950i)T \)
5 \( 1 + (-0.142 + 0.989i)T \)
7 \( 1 + (0.786 + 0.618i)T \)
11 \( 1 + (-0.786 + 0.618i)T \)
17 \( 1 + (0.0475 + 0.998i)T \)
19 \( 1 + (-0.981 + 0.189i)T \)
23 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (-0.928 + 0.371i)T \)
31 \( 1 + (0.654 - 0.755i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.580 + 0.814i)T \)
43 \( 1 + (-0.928 - 0.371i)T \)
47 \( 1 + (0.415 + 0.909i)T \)
53 \( 1 + (0.415 - 0.909i)T \)
59 \( 1 + (0.995 + 0.0950i)T \)
61 \( 1 + (-0.235 + 0.971i)T \)
67 \( 1 + (-0.995 + 0.0950i)T \)
71 \( 1 + (0.928 + 0.371i)T \)
73 \( 1 + (-0.654 + 0.755i)T \)
79 \( 1 + (-0.654 + 0.755i)T \)
83 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (-0.786 - 0.618i)T \)
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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

−21.20658912149411089880463609076, −20.52525837860800799301982559904, −19.66311231373537662767108636331, −18.850706905329429308154911778003, −18.03234003746005818524729833091, −17.15062150026760910626187717855, −16.4578081534056125688450539792, −15.70338343598478057264896472320, −15.10300764846618154218074203984, −14.18424913522662970386146163584, −13.429521361744786258644447022543, −13.1775335888880191455146548646, −11.98889066103179961860580126385, −10.75151939300497394198415875559, −9.76978930179334035829818990077, −8.92091424126475269516254099069, −8.36763303272538852611698375029, −7.67286933325899745173606570271, −7.05852842278006332780226868894, −5.60822418955401597568943961030, −4.87570837598829032316272685415, −4.1933742889016401180630497920, −3.27153465712696356317622172524, −1.83154287102322261763243288235, −0.615894852514555619863017844, 1.574263471587052412610069653624, 2.32581892246920701319813594666, 2.86329904926233575031711650278, 3.97024343303097688637716689055, 4.65036401398414932164880412569, 5.888539917450262219420588589770, 7.07634824870989406976597003534, 8.26324811361303652373855702215, 8.35360595160609012319915472272, 9.67637882595264110580456088032, 10.31522267000054422291621211514, 11.006226950084661316923021469595, 11.9396255148937086982556934459, 12.83086780989361897202649903936, 13.38289999764689256308885895215, 14.58522804351980551087214758197, 14.803698831363965843553203622891, 15.401947656904461099531997409053, 17.00688812733421231091997899766, 17.97948109870322667869477824919, 18.58731831657802194736900863508, 18.949892429530027077518113844477, 19.83615608912148884505341685833, 20.68523625146913006144100710574, 21.18216300456972089697380749724

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