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Properties

Label 2-1012-253.142-c0-0-0
Degree $2$
Conductor $1012$
Sign $0.740 - 0.671i$
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  + (−1.78 + 0.523i)3-s + (0.975 + 0.627i)5-s + (2.05 − 1.32i)9-s + (0.415 − 0.909i)11-s + (−2.06 − 0.606i)15-s + (0.580 + 0.814i)23-s + (0.143 + 0.314i)25-s + (−1.75 + 2.03i)27-s + (1.70 + 0.500i)31-s + (−0.264 + 1.83i)33-s + (−0.550 + 0.353i)37-s + 2.83·45-s + 1.68·47-s + (−0.959 + 0.281i)49-s + (0.186 + 1.29i)53-s + ⋯
L(s)  = 1  + (−1.78 + 0.523i)3-s + (0.975 + 0.627i)5-s + (2.05 − 1.32i)9-s + (0.415 − 0.909i)11-s + (−2.06 − 0.606i)15-s + (0.580 + 0.814i)23-s + (0.143 + 0.314i)25-s + (−1.75 + 2.03i)27-s + (1.70 + 0.500i)31-s + (−0.264 + 1.83i)33-s + (−0.550 + 0.353i)37-s + 2.83·45-s + 1.68·47-s + (−0.959 + 0.281i)49-s + (0.186 + 1.29i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $0.740 - 0.671i$
Analytic conductor: \(0.505053\)
Root analytic conductor: \(0.710671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1012,\ (\ :0),\ 0.740 - 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7235918170\)
\(L(\frac12)\) \(\approx\) \(0.7235918170\)
\(L(1)\) 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 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (-0.580 - 0.814i)T \)
good3 \( 1 + (1.78 - 0.523i)T + (0.841 - 0.540i)T^{2} \)
5 \( 1 + (-0.975 - 0.627i)T + (0.415 + 0.909i)T^{2} \)
7 \( 1 + (0.959 - 0.281i)T^{2} \)
13 \( 1 + (0.959 + 0.281i)T^{2} \)
17 \( 1 + (0.142 - 0.989i)T^{2} \)
19 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.142 - 0.989i)T^{2} \)
31 \( 1 + (-1.70 - 0.500i)T + (0.841 + 0.540i)T^{2} \)
37 \( 1 + (0.550 - 0.353i)T + (0.415 - 0.909i)T^{2} \)
41 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (-0.841 + 0.540i)T^{2} \)
47 \( 1 - 1.68T + T^{2} \)
53 \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \)
59 \( 1 + (0.279 - 1.94i)T + (-0.959 - 0.281i)T^{2} \)
61 \( 1 + (-0.841 - 0.540i)T^{2} \)
67 \( 1 + (0.653 + 1.43i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (0.827 + 1.81i)T + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.959 + 0.281i)T^{2} \)
83 \( 1 + (-0.415 + 0.909i)T^{2} \)
89 \( 1 + (-1.50 + 0.442i)T + (0.841 - 0.540i)T^{2} \)
97 \( 1 + (1.49 + 0.961i)T + (0.415 + 0.909i)T^{2} \)
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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

−10.59136349681379346336062596791, −9.694330985515113643573226046886, −8.936981514896942291508407334038, −7.42383167740252787014917971704, −6.40477696759621647563543152999, −6.08666719589467035632042644317, −5.27411461096678353176741274900, −4.33185455203400020049958875029, −3.06049570239984864339456134687, −1.26148405251957857343287873247, 1.09158849883351725535540448693, 2.15661043249775789286381998682, 4.34110563688094718670185024805, 5.04525955617853704353805791920, 5.77324584614099675641224288504, 6.56948807860227876452804712691, 7.15384782205572984097468035250, 8.376682684691236663599242939515, 9.565553246113512161219827611680, 10.10003951094501347049259503176

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