Properties

Label 2-1012-253.164-c0-0-1
Degree $2$
Conductor $1012$
Sign $0.914 + 0.405i$
Analytic cond. $0.505053$
Root an. cond. $0.710671$
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  + (0.0395 − 0.0865i)3-s + (1.02 − 1.18i)5-s + (0.648 + 0.748i)9-s + (−0.142 + 0.989i)11-s + (−0.0621 − 0.136i)15-s + (−0.786 − 0.618i)23-s + (−0.209 − 1.45i)25-s + (0.181 − 0.0533i)27-s + (−0.271 − 0.595i)31-s + (0.0800 + 0.0514i)33-s + (−0.308 − 0.356i)37-s + 1.55·45-s − 1.30·47-s + (0.415 − 0.909i)49-s + (−1.61 + 1.03i)53-s + ⋯
L(s)  = 1  + (0.0395 − 0.0865i)3-s + (1.02 − 1.18i)5-s + (0.648 + 0.748i)9-s + (−0.142 + 0.989i)11-s + (−0.0621 − 0.136i)15-s + (−0.786 − 0.618i)23-s + (−0.209 − 1.45i)25-s + (0.181 − 0.0533i)27-s + (−0.271 − 0.595i)31-s + (0.0800 + 0.0514i)33-s + (−0.308 − 0.356i)37-s + 1.55·45-s − 1.30·47-s + (0.415 − 0.909i)49-s + (−1.61 + 1.03i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $0.914 + 0.405i$
Analytic conductor: \(0.505053\)
Root analytic conductor: \(0.710671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1012,\ (\ :0),\ 0.914 + 0.405i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.244130132\)
\(L(\frac12)\) \(\approx\) \(1.244130132\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (0.786 + 0.618i)T \)
good3 \( 1 + (-0.0395 + 0.0865i)T + (-0.654 - 0.755i)T^{2} \)
5 \( 1 + (-1.02 + 1.18i)T + (-0.142 - 0.989i)T^{2} \)
7 \( 1 + (-0.415 + 0.909i)T^{2} \)
13 \( 1 + (-0.415 - 0.909i)T^{2} \)
17 \( 1 + (-0.841 - 0.540i)T^{2} \)
19 \( 1 + (-0.841 + 0.540i)T^{2} \)
29 \( 1 + (-0.841 - 0.540i)T^{2} \)
31 \( 1 + (0.271 + 0.595i)T + (-0.654 + 0.755i)T^{2} \)
37 \( 1 + (0.308 + 0.356i)T + (-0.142 + 0.989i)T^{2} \)
41 \( 1 + (0.142 + 0.989i)T^{2} \)
43 \( 1 + (0.654 + 0.755i)T^{2} \)
47 \( 1 + 1.30T + T^{2} \)
53 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
59 \( 1 + (-1.21 - 0.782i)T + (0.415 + 0.909i)T^{2} \)
61 \( 1 + (0.654 - 0.755i)T^{2} \)
67 \( 1 + (-0.252 - 1.75i)T + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (0.264 + 1.83i)T + (-0.959 + 0.281i)T^{2} \)
73 \( 1 + (-0.841 + 0.540i)T^{2} \)
79 \( 1 + (-0.415 - 0.909i)T^{2} \)
83 \( 1 + (0.142 - 0.989i)T^{2} \)
89 \( 1 + (0.738 - 1.61i)T + (-0.654 - 0.755i)T^{2} \)
97 \( 1 + (-0.428 + 0.494i)T + (-0.142 - 0.989i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.954042647139543944238467811270, −9.425163999095514131142688888060, −8.482866965185489347417032650232, −7.70134231356421130219300221180, −6.71316367292014730002898708508, −5.66040370042887656587442794899, −4.89930911159952235939783786083, −4.18533566456332004062009129437, −2.33354316913659955937506022092, −1.54837841708496927171132103802, 1.68900360270520557452273090533, 2.97957961096956897971060052517, 3.72025200709799855522063299147, 5.18196656051759068447656997494, 6.18453168939110037622514999858, 6.61024557115229771251440369293, 7.60330428610133391336156673239, 8.666543542864839743999107647519, 9.670737357570697547751946417764, 10.06084890035109153060006252313

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