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Properties

Label 2-1012-253.197-c0-0-0
Degree $2$
Conductor $1012$
Sign $0.360 - 0.932i$
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  + (1.21 + 0.782i)3-s + (−0.271 + 0.595i)5-s + (0.454 + 0.996i)9-s + (−0.654 + 0.755i)11-s + (−0.796 + 0.511i)15-s + (−0.327 − 0.945i)23-s + (0.374 + 0.432i)25-s + (−0.0196 + 0.136i)27-s + (0.975 − 0.627i)31-s + (−1.38 + 0.407i)33-s + (−0.653 − 1.43i)37-s − 0.716·45-s + 0.830·47-s + (0.841 + 0.540i)49-s + (0.273 + 0.0801i)53-s + ⋯
L(s)  = 1  + (1.21 + 0.782i)3-s + (−0.271 + 0.595i)5-s + (0.454 + 0.996i)9-s + (−0.654 + 0.755i)11-s + (−0.796 + 0.511i)15-s + (−0.327 − 0.945i)23-s + (0.374 + 0.432i)25-s + (−0.0196 + 0.136i)27-s + (0.975 − 0.627i)31-s + (−1.38 + 0.407i)33-s + (−0.653 − 1.43i)37-s − 0.716·45-s + 0.830·47-s + (0.841 + 0.540i)49-s + (0.273 + 0.0801i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $0.360 - 0.932i$
Analytic conductor: \(0.505053\)
Root analytic conductor: \(0.710671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1012,\ (\ :0),\ 0.360 - 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.382723449\)
\(L(\frac12)\) \(\approx\) \(1.382723449\)
\(L(1)\) 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 \( 1 + (0.654 - 0.755i)T \)
23 \( 1 + (0.327 + 0.945i)T \)
good3 \( 1 + (-1.21 - 0.782i)T + (0.415 + 0.909i)T^{2} \)
5 \( 1 + (0.271 - 0.595i)T + (-0.654 - 0.755i)T^{2} \)
7 \( 1 + (-0.841 - 0.540i)T^{2} \)
13 \( 1 + (-0.841 + 0.540i)T^{2} \)
17 \( 1 + (0.959 - 0.281i)T^{2} \)
19 \( 1 + (0.959 + 0.281i)T^{2} \)
29 \( 1 + (0.959 - 0.281i)T^{2} \)
31 \( 1 + (-0.975 + 0.627i)T + (0.415 - 0.909i)T^{2} \)
37 \( 1 + (0.653 + 1.43i)T + (-0.654 + 0.755i)T^{2} \)
41 \( 1 + (0.654 + 0.755i)T^{2} \)
43 \( 1 + (-0.415 - 0.909i)T^{2} \)
47 \( 1 - 0.830T + T^{2} \)
53 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
59 \( 1 + (1.78 - 0.523i)T + (0.841 - 0.540i)T^{2} \)
61 \( 1 + (-0.415 + 0.909i)T^{2} \)
67 \( 1 + (0.308 + 0.356i)T + (-0.142 + 0.989i)T^{2} \)
71 \( 1 + (1.28 + 1.48i)T + (-0.142 + 0.989i)T^{2} \)
73 \( 1 + (0.959 + 0.281i)T^{2} \)
79 \( 1 + (-0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.654 - 0.755i)T^{2} \)
89 \( 1 + (-0.396 - 0.254i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 + (-0.481 + 1.05i)T + (-0.654 - 0.755i)T^{2} \)
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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

−10.41997574565690327932891474869, −9.392105165099058423391970892660, −8.808489122915485582557569336634, −7.83151606969805791198408860212, −7.28687507842964071426842747401, −6.08864353549397128471539686966, −4.78113736270626504023435164294, −4.02861122652029554529322039160, −3.01644498551699686663780922590, −2.26130740103019900343500821850, 1.31604613221269916961202128944, 2.62924094421533268609899014411, 3.43722614406644270914969826453, 4.67332606959933596149743614868, 5.75385632401435953609409733255, 6.88183670898327253332883471057, 7.71050923026699684680481488557, 8.420963418030121804796378652227, 8.776089377213715689074792042059, 9.840504694512885674310128899269

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