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Properties

Label 1-1012-1012.351-r0-0-0
Degree $1$
Conductor $1012$
Sign $0.952 + 0.305i$
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  + (−0.841 − 0.540i)3-s + (0.415 − 0.909i)5-s + (−0.959 − 0.281i)7-s + (0.415 + 0.909i)9-s + (0.959 − 0.281i)13-s + (−0.841 + 0.540i)15-s + (0.142 + 0.989i)17-s + (−0.142 + 0.989i)19-s + (0.654 + 0.755i)21-s + (−0.654 − 0.755i)25-s + (0.142 − 0.989i)27-s + (0.142 + 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.654 + 0.755i)35-s + (0.415 + 0.909i)37-s + ⋯
L(s)  = 1  + (−0.841 − 0.540i)3-s + (0.415 − 0.909i)5-s + (−0.959 − 0.281i)7-s + (0.415 + 0.909i)9-s + (0.959 − 0.281i)13-s + (−0.841 + 0.540i)15-s + (0.142 + 0.989i)17-s + (−0.142 + 0.989i)19-s + (0.654 + 0.755i)21-s + (−0.654 − 0.755i)25-s + (0.142 − 0.989i)27-s + (0.142 + 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.654 + 0.755i)35-s + (0.415 + 0.909i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $0.952 + 0.305i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ 0.952 + 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8672338346 + 0.1358007980i\)
\(L(\frac12)\) \(\approx\) \(0.8672338346 + 0.1358007980i\)
\(L(1)\) \(\approx\) \(0.7806899103 - 0.1268454877i\)
\(L(1)\) \(\approx\) \(0.7806899103 - 0.1268454877i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.841 - 0.540i)T \)
5 \( 1 + (0.415 - 0.909i)T \)
7 \( 1 + (-0.959 - 0.281i)T \)
13 \( 1 + (0.959 - 0.281i)T \)
17 \( 1 + (0.142 + 0.989i)T \)
19 \( 1 + (-0.142 + 0.989i)T \)
29 \( 1 + (0.142 + 0.989i)T \)
31 \( 1 + (-0.841 + 0.540i)T \)
37 \( 1 + (0.415 + 0.909i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.959 - 0.281i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
71 \( 1 + (0.654 + 0.755i)T \)
73 \( 1 + (0.142 - 0.989i)T \)
79 \( 1 + (-0.959 + 0.281i)T \)
83 \( 1 + (0.415 + 0.909i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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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

−21.67967449919779422959913416469, −21.09236499242553687456306593133, −20.10245864793479044843893673975, −18.9551951884050518026449902150, −18.47017325614033908266754652478, −17.6819668095598363922105690333, −16.90243939983552616074064567528, −15.878162292712356537449968213453, −15.633873523678994663841257908476, −14.56626248997881168122772165420, −13.60993473194723539598982338145, −12.87957892186670911480954546261, −11.78208681813239157314409595768, −11.13670779062900561108312789774, −10.43469122021021838978106117244, −9.50047714196150835408018604997, −9.09513385520178475345116990436, −7.446859575658258735908566209476, −6.59825779217681613371658906117, −6.07011464045493807433560033748, −5.22308286091527247266859812012, −3.99548388617802163413982913259, −3.22036982306259339008853725398, −2.18056462719633415584183076473, −0.49258562430705202093525126024, 1.05589295002043231562953804569, 1.728178457568820788191877475467, 3.28017589435807472176286354675, 4.29005676578771725353974519242, 5.37153255878455497829777859427, 6.0798168392748387164966531714, 6.6548271007628117761294704285, 7.893543899560227587857303114732, 8.60347422328404769029816008571, 9.751595158127734190129274025298, 10.41283622274876406884358358181, 11.29063663897518718736184698805, 12.436585575412274982674834345568, 12.81224361857235591518614191985, 13.38907736466285970239334038649, 14.393788798620253757017086123355, 15.73293046643300564095115287312, 16.44113106897063949986141538838, 16.78971923931449608781952258189, 17.74562478531951267117672434658, 18.426668000971328014994934013226, 19.31362383811435575008292560932, 20.02992664632956397956385155465, 20.92479031510482450007598258909, 21.76463530309913023306805566954

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