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Properties

Label 1-1012-1012.151-r0-0-0
Degree $1$
Conductor $1012$
Sign $0.995 - 0.0899i$
Analytic cond. $4.69970$
Root an. cond. $4.69970$
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  + (0.870 + 0.491i)3-s + (0.516 − 0.856i)5-s + (−0.362 − 0.931i)7-s + (0.516 + 0.856i)9-s + (0.254 + 0.967i)13-s + (0.870 − 0.491i)15-s + (0.564 + 0.825i)17-s + (0.941 − 0.336i)19-s + (0.142 − 0.989i)21-s + (−0.466 − 0.884i)25-s + (0.0285 + 0.999i)27-s + (−0.941 − 0.336i)29-s + (0.736 + 0.676i)31-s + (−0.985 − 0.170i)35-s + (0.974 − 0.226i)37-s + ⋯
L(s)  = 1  + (0.870 + 0.491i)3-s + (0.516 − 0.856i)5-s + (−0.362 − 0.931i)7-s + (0.516 + 0.856i)9-s + (0.254 + 0.967i)13-s + (0.870 − 0.491i)15-s + (0.564 + 0.825i)17-s + (0.941 − 0.336i)19-s + (0.142 − 0.989i)21-s + (−0.466 − 0.884i)25-s + (0.0285 + 0.999i)27-s + (−0.941 − 0.336i)29-s + (0.736 + 0.676i)31-s + (−0.985 − 0.170i)35-s + (0.974 − 0.226i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $0.995 - 0.0899i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ 0.995 - 0.0899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.338636483 - 0.1053678743i\)
\(L(\frac12)\) \(\approx\) \(2.338636483 - 0.1053678743i\)
\(L(1)\) \(\approx\) \(1.577608697 + 0.01649492246i\)
\(L(1)\) \(\approx\) \(1.577608697 + 0.01649492246i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.870 + 0.491i)T \)
5 \( 1 + (0.516 - 0.856i)T \)
7 \( 1 + (-0.362 - 0.931i)T \)
13 \( 1 + (0.254 + 0.967i)T \)
17 \( 1 + (0.564 + 0.825i)T \)
19 \( 1 + (0.941 - 0.336i)T \)
29 \( 1 + (-0.941 - 0.336i)T \)
31 \( 1 + (0.736 + 0.676i)T \)
37 \( 1 + (0.974 - 0.226i)T \)
41 \( 1 + (-0.974 - 0.226i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.774 - 0.633i)T \)
59 \( 1 + (0.998 + 0.0570i)T \)
61 \( 1 + (-0.993 - 0.113i)T \)
67 \( 1 + (0.142 - 0.989i)T \)
71 \( 1 + (0.985 - 0.170i)T \)
73 \( 1 + (-0.610 - 0.791i)T \)
79 \( 1 + (-0.254 - 0.967i)T \)
83 \( 1 + (-0.921 - 0.389i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (-0.921 + 0.389i)T \)
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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

−21.64938301389231466642138627523, −20.718077125209107448941472943802, −20.16425827985855389838324793328, −19.07555359215654644235965269009, −18.42923362346418139228607060774, −18.221697499962263564902873978903, −17.0898885280332807653466878804, −15.820870558763460094081299718776, −15.2360410809345679615656243317, −14.51153963983613524444170261148, −13.73549625723380975257888001463, −13.06465243086545469734170434964, −12.17544331977196838591509393242, −11.360174565354786181957100464378, −10.07003376352676064643142338373, −9.61276559708755827559287944053, −8.69862810863224504197393916650, −7.73021258179927451224275216732, −7.07035626592357329959400826547, −6.01002225060365632150909920867, −5.42388940559844110874737658079, −3.73884575882936741180709258532, −2.90006767554098728748696416619, −2.43942331233623333384769244206, −1.162055405225996520168640950807, 1.11914355406834266286564633579, 2.02585941022770573871202265922, 3.30099904719403965784540658947, 4.09297881144894775624428702394, 4.83368502759800701951632479482, 5.92392354605669806040123921571, 7.04552520391862675523197293087, 7.93020095187669389041101159011, 8.77819884537975941343202882052, 9.564359176937048852225092325671, 10.06564353278271259580893852339, 11.04761985760918027874787576207, 12.19886139595993106772680535886, 13.2113003419274673097148744434, 13.6592520804260185210315362758, 14.34091636947798118967583163987, 15.32606792429726519397832711993, 16.38653476820191102412446278149, 16.57625104925341778052852803339, 17.5487854123096535457236365729, 18.67249423829588680530301149683, 19.54259087759028851478265698409, 20.09126416090392954380541624284, 20.842289548667720681996400042303, 21.38390520889806898219685585765

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