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Properties

Label 2-154-77.76-c1-0-3
Degree $2$
Conductor $154$
Sign $0.983 - 0.181i$
Analytic cond. $1.22969$
Root an. cond. $1.10891$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + 0.646i·3-s − 4-s + 3.09i·5-s + 0.646·6-s + (2.44 + i)7-s + i·8-s + 2.58·9-s + 3.09·10-s + (−1.79 − 2.79i)11-s − 0.646i·12-s + 0.646·13-s + (1 − 2.44i)14-s − 2·15-s + 16-s − 3.74·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.373i·3-s − 0.5·4-s + 1.38i·5-s + 0.263·6-s + (0.925 + 0.377i)7-s + 0.353i·8-s + 0.860·9-s + 0.978·10-s + (−0.540 − 0.841i)11-s − 0.186i·12-s + 0.179·13-s + (0.267 − 0.654i)14-s − 0.516·15-s + 0.250·16-s − 0.907·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $0.983 - 0.181i$
Analytic conductor: \(1.22969\)
Root analytic conductor: \(1.10891\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{154} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ 0.983 - 0.181i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15432 + 0.105888i\)
\(L(\frac12)\) \(\approx\) \(1.15432 + 0.105888i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
7 \( 1 + (-2.44 - i)T \)
11 \( 1 + (1.79 + 2.79i)T \)
good3 \( 1 - 0.646iT - 3T^{2} \)
5 \( 1 - 3.09iT - 5T^{2} \)
13 \( 1 - 0.646T + 13T^{2} \)
17 \( 1 + 3.74T + 17T^{2} \)
19 \( 1 + 1.80T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 1.58iT - 29T^{2} \)
31 \( 1 + 8.64iT - 31T^{2} \)
37 \( 1 - 3.58T + 37T^{2} \)
41 \( 1 + 9.93T + 41T^{2} \)
43 \( 1 + 7.16iT - 43T^{2} \)
47 \( 1 + 9.93iT - 47T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 + 0.646iT - 59T^{2} \)
61 \( 1 - 1.93T + 61T^{2} \)
67 \( 1 + 7.58T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 16.1T + 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 - 9.79iT - 89T^{2} \)
97 \( 1 + 8.50iT - 97T^{2} \)
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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

−13.05317034488487921291112697284, −11.61162652491996453867253879362, −10.91522135798017796149844150165, −10.36320578213068537174531811652, −9.062927822856397291938575553543, −7.87374893120981154195912390966, −6.57556235341366601016707768974, −5.04672329369163560464278041221, −3.66316073096595015160724651726, −2.27324442557031176208034093767, 1.46982430814631866928688820913, 4.50850069932062857449365930657, 4.94785102320726863390187966568, 6.68003536146994254660274295545, 7.74466508032361560885412356963, 8.550580993872883550579195086633, 9.623370883429531126852752764564, 10.90650075996229289705702472651, 12.37957784457156496753764325284, 12.94945010765454213820518266708

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