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Properties

Label 1-151-151.37-r0-0-0
Degree $1$
Conductor $151$
Sign $0.996 + 0.0838i$
Analytic cond. $0.701241$
Root an. cond. $0.701241$
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  + (−0.978 + 0.207i)2-s + (−0.637 − 0.770i)3-s + (0.913 − 0.406i)4-s + (0.944 − 0.328i)5-s + (0.783 + 0.621i)6-s + (0.604 + 0.796i)7-s + (−0.809 + 0.587i)8-s + (−0.187 + 0.982i)9-s + (−0.855 + 0.518i)10-s + (0.985 + 0.166i)11-s + (−0.895 − 0.444i)12-s + (−0.699 + 0.714i)13-s + (−0.756 − 0.653i)14-s + (−0.855 − 0.518i)15-s + (0.669 − 0.743i)16-s + (0.387 + 0.921i)17-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)2-s + (−0.637 − 0.770i)3-s + (0.913 − 0.406i)4-s + (0.944 − 0.328i)5-s + (0.783 + 0.621i)6-s + (0.604 + 0.796i)7-s + (−0.809 + 0.587i)8-s + (−0.187 + 0.982i)9-s + (−0.855 + 0.518i)10-s + (0.985 + 0.166i)11-s + (−0.895 − 0.444i)12-s + (−0.699 + 0.714i)13-s + (−0.756 − 0.653i)14-s + (−0.855 − 0.518i)15-s + (0.669 − 0.743i)16-s + (0.387 + 0.921i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.996 + 0.0838i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.996 + 0.0838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7393071006 + 0.03103948158i\)
\(L(\frac12)\) \(\approx\) \(0.7393071006 + 0.03103948158i\)
\(L(1)\) \(\approx\) \(0.7330231120 + 0.01592743239i\)
\(L(1)\) \(\approx\) \(0.7330231120 + 0.01592743239i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (-0.978 + 0.207i)T \)
3 \( 1 + (-0.637 - 0.770i)T \)
5 \( 1 + (0.944 - 0.328i)T \)
7 \( 1 + (0.604 + 0.796i)T \)
11 \( 1 + (0.985 + 0.166i)T \)
13 \( 1 + (-0.699 + 0.714i)T \)
17 \( 1 + (0.387 + 0.921i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.104 + 0.994i)T \)
29 \( 1 + (0.535 + 0.844i)T \)
31 \( 1 + (0.996 - 0.0836i)T \)
37 \( 1 + (-0.268 - 0.963i)T \)
41 \( 1 + (-0.929 + 0.368i)T \)
43 \( 1 + (0.604 - 0.796i)T \)
47 \( 1 + (0.146 - 0.989i)T \)
53 \( 1 + (0.0627 - 0.998i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (-0.348 - 0.937i)T \)
67 \( 1 + (-0.187 - 0.982i)T \)
71 \( 1 + (0.387 - 0.921i)T \)
73 \( 1 + (-0.992 + 0.125i)T \)
79 \( 1 + (-0.425 - 0.904i)T \)
83 \( 1 + (0.728 + 0.684i)T \)
89 \( 1 + (-0.957 + 0.289i)T \)
97 \( 1 + (-0.756 + 0.653i)T \)
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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

−27.809382563562976092234273668382, −27.143535981840370575424551922550, −26.48430166603185889053010742278, −25.277559599285595325260828059146, −24.45458064277090627780564758836, −22.89555219769677896953243130965, −21.98634098899238916744126467913, −20.93998891550204275751800656259, −20.38349292715399665784631929673, −18.95587173117019316557852714609, −17.689327159152595988989222154017, −17.224426653585462449413633964454, −16.49373137395846128619193830071, −15.06045319537558196488582348632, −14.11351521600848862327523141434, −12.329201565374632116596078772650, −11.29161412774195541873408580197, −10.26203609018235992990675072763, −9.816905084331866053868220227863, −8.471346224643072758931220702726, −6.96780049507282624247595612732, −5.9882189615443533025890450012, −4.44824396739024559067804362126, −2.84903611025673977837801405221, −1.103265776449733351133391809457, 1.48049172988102795669434405447, 2.19015374356054166970417664603, 5.06032787011329234078802073198, 6.10043071367410935370372342077, 6.95294624574542290706576436092, 8.3698936849143550267256821278, 9.25057644539613271671333092782, 10.50014441110881233404633315421, 11.716877073142048226093895792110, 12.419229443084017030851894924059, 13.98877494219034086231640122141, 15.02969693785991703609483194970, 16.57801100026785522666391492173, 17.336503982154221931714830970531, 17.81831041540139539529250808204, 19.000785516520864869742193993207, 19.710398613642354480865738963583, 21.30539351439634297998439081344, 21.91966820673271807288227598257, 23.64847029611542722431994351635, 24.40767502016488690410927278040, 25.1165110622826325502755507655, 25.85955278238954575781126022549, 27.42861643490167193240305034646, 28.140438928498751202851643657633

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