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Properties

Label 2-154-11.5-c1-0-2
Degree $2$
Conductor $154$
Sign $0.944 - 0.329i$
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  + (−0.309 − 0.951i)2-s + (1.80 + 1.31i)3-s + (−0.809 + 0.587i)4-s + (−0.809 + 2.48i)5-s + (0.690 − 2.12i)6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (0.618 + 1.90i)9-s + 2.61·10-s + (1.23 + 3.07i)11-s − 2.23·12-s + (−0.927 − 2.85i)13-s + (−0.809 − 0.587i)14-s + (−4.73 + 3.44i)15-s + (0.309 − 0.951i)16-s + (1.69 − 5.20i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (1.04 + 0.758i)3-s + (−0.404 + 0.293i)4-s + (−0.361 + 1.11i)5-s + (0.282 − 0.868i)6-s + (0.305 − 0.222i)7-s + (0.286 + 0.207i)8-s + (0.206 + 0.634i)9-s + 0.827·10-s + (0.372 + 0.927i)11-s − 0.645·12-s + (−0.257 − 0.791i)13-s + (−0.216 − 0.157i)14-s + (−1.22 + 0.888i)15-s + (0.0772 − 0.237i)16-s + (0.410 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $0.944 - 0.329i$
Analytic conductor: \(1.22969\)
Root analytic conductor: \(1.10891\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{154} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ 0.944 - 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25618 + 0.213140i\)
\(L(\frac12)\) \(\approx\) \(1.25618 + 0.213140i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-1.23 - 3.07i)T \)
good3 \( 1 + (-1.80 - 1.31i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.809 - 2.48i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (0.927 + 2.85i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.69 + 5.20i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.80 + 2.04i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 + (2.42 - 1.76i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.69 + 8.28i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (8.35 - 6.06i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-8.47 - 6.15i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.85T + 43T^{2} \)
47 \( 1 + (6.35 + 4.61i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.16 - 3.57i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.5 - 3.26i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.972 - 2.99i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 13T + 67T^{2} \)
71 \( 1 + (0.0729 - 0.224i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.92 + 5.03i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.39 + 7.38i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.42 - 4.39i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 1.85T + 89T^{2} \)
97 \( 1 + (-0.145 - 0.449i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.07351156616393586351527693432, −11.77071540446740782143135003872, −10.84938312832261443195747148187, −9.946917167594023471484503603669, −9.214075921333835001834213746821, −7.941953944939342920621626178680, −7.01421498252725234663065818171, −4.77523263836172278735038831932, −3.52743895421427828788094269508, −2.58837258557473012551456344410, 1.61156591368215639983201813971, 3.80728351752043757670000531566, 5.32149998824142016568908187618, 6.73949629244864289013857599219, 7.980968530535992841072680640921, 8.587385263427838896207430832725, 9.162209892238880285707230362696, 10.88908227481295728350106470835, 12.37735983730944499691841156813, 12.92331357325388732163837258329

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