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Properties

Label 1-1157-1157.671-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.518 - 0.855i$
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  + (0.909 + 0.415i)2-s + (−0.0713 − 0.997i)3-s + (0.654 + 0.755i)4-s + (0.959 − 0.281i)5-s + (0.349 − 0.936i)6-s + (−0.877 − 0.479i)7-s + (0.281 + 0.959i)8-s + (−0.989 + 0.142i)9-s + (0.989 + 0.142i)10-s + (0.281 − 0.959i)11-s + (0.707 − 0.707i)12-s + (−0.599 − 0.800i)14-s + (−0.349 − 0.936i)15-s + (−0.142 + 0.989i)16-s + (0.909 − 0.415i)17-s + (−0.959 − 0.281i)18-s + ⋯
L(s)  = 1  + (0.909 + 0.415i)2-s + (−0.0713 − 0.997i)3-s + (0.654 + 0.755i)4-s + (0.959 − 0.281i)5-s + (0.349 − 0.936i)6-s + (−0.877 − 0.479i)7-s + (0.281 + 0.959i)8-s + (−0.989 + 0.142i)9-s + (0.989 + 0.142i)10-s + (0.281 − 0.959i)11-s + (0.707 − 0.707i)12-s + (−0.599 − 0.800i)14-s + (−0.349 − 0.936i)15-s + (−0.142 + 0.989i)16-s + (0.909 − 0.415i)17-s + (−0.959 − 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.518 - 0.855i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.518 - 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.471223271 - 1.392019036i\)
\(L(\frac12)\) \(\approx\) \(2.471223271 - 1.392019036i\)
\(L(1)\) \(\approx\) \(1.850125023 - 0.4031555317i\)
\(L(1)\) \(\approx\) \(1.850125023 - 0.4031555317i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.909 + 0.415i)T \)
3 \( 1 + (-0.0713 - 0.997i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (-0.877 - 0.479i)T \)
11 \( 1 + (0.281 - 0.959i)T \)
17 \( 1 + (0.909 - 0.415i)T \)
19 \( 1 + (0.800 + 0.599i)T \)
23 \( 1 + (-0.800 - 0.599i)T \)
29 \( 1 + (-0.877 - 0.479i)T \)
31 \( 1 + (0.599 + 0.800i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (0.0713 - 0.997i)T \)
43 \( 1 + (0.877 - 0.479i)T \)
47 \( 1 + (-0.654 - 0.755i)T \)
53 \( 1 + (-0.755 - 0.654i)T \)
59 \( 1 + (-0.997 - 0.0713i)T \)
61 \( 1 + (0.977 - 0.212i)T \)
67 \( 1 + (-0.755 - 0.654i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (-0.989 - 0.142i)T \)
79 \( 1 + (0.989 + 0.142i)T \)
83 \( 1 + (-0.349 + 0.936i)T \)
97 \( 1 + (-0.281 - 0.959i)T \)
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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

−21.622791803762353604421878551064, −20.761411369470900873877166086852, −20.17696720983902574598293489291, −19.3642373781359513302493786706, −18.40382080602098482392534885156, −17.44837162524462010873676054049, −16.54153506881476911393188912004, −15.828723873599197483796353169886, −15.0006573711900120995721425824, −14.52279173758295039462626227292, −13.61240253555282268424504981032, −12.85302847387740797933921447317, −11.994179842955673783767737312539, −11.21681630829357678173325661556, −10.20941670962204328889529659194, −9.63678396366169798740897517098, −9.34428545744418725643590734009, −7.615111127770497267141526761942, −6.391489158780153941638120882027, −5.8937228613983392742690251630, −5.114332277554680292128069963457, −4.19830726744911377226477554646, −3.19061148491524742430011597872, −2.62990650501555159252094249554, −1.45671702642754145553237299897, 0.8609794414667503256488576733, 2.03166780733175341447863960506, 3.00241785058998268527685553936, 3.75858658710881579269847449055, 5.21509845282277640660331111624, 5.90721846067099912993304536885, 6.37908616603687659271873187444, 7.2925200333566766358274009996, 8.08207772293581950947440577237, 9.07524901715380007974973268881, 10.123078051693035189837420432082, 11.1565970251360908352794204174, 12.15460744579886019861240644643, 12.64601365696684804361384829768, 13.50035923130148530556310552353, 14.000566641952647167611883954, 14.411257039889312669182196663229, 15.95779196008701863368232998782, 16.55160859806666591519203059167, 17.04771972706989509786649928590, 17.98645120824542208337404028326, 18.82035148780114653182610926712, 19.71028876496840882821885080959, 20.51215297818496173717422049791, 21.175841085066233924454701807874

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