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Properties

Label 2-1175-5.4-c1-0-1
Degree $2$
Conductor $1175$
Sign $0.447 + 0.894i$
Analytic cond. $9.38242$
Root an. cond. $3.06307$
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  + 1.41i·2-s + 3.19i·3-s − 0.0122·4-s − 4.52·6-s − 4.71i·7-s + 2.81i·8-s − 7.17·9-s − 5.53·11-s − 0.0389i·12-s − 0.648i·13-s + 6.68·14-s − 4.02·16-s + 1.05i·17-s − 10.1i·18-s − 7.88·19-s + ⋯
L(s)  = 1  + 1.00i·2-s + 1.84i·3-s − 0.00611·4-s − 1.84·6-s − 1.78i·7-s + 0.996i·8-s − 2.39·9-s − 1.66·11-s − 0.0112i·12-s − 0.179i·13-s + 1.78·14-s − 1.00·16-s + 0.256i·17-s − 2.40i·18-s − 1.80·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1175\)    =    \(5^{2} \cdot 47\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(9.38242\)
Root analytic conductor: \(3.06307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1175} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1175,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3672626138\)
\(L(\frac12)\) \(\approx\) \(0.3672626138\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
47 \( 1 + iT \)
good2 \( 1 - 1.41iT - 2T^{2} \)
3 \( 1 - 3.19iT - 3T^{2} \)
7 \( 1 + 4.71iT - 7T^{2} \)
11 \( 1 + 5.53T + 11T^{2} \)
13 \( 1 + 0.648iT - 13T^{2} \)
17 \( 1 - 1.05iT - 17T^{2} \)
19 \( 1 + 7.88T + 19T^{2} \)
23 \( 1 - 3.05iT - 23T^{2} \)
29 \( 1 - 3.58T + 29T^{2} \)
31 \( 1 - 4.18T + 31T^{2} \)
37 \( 1 - 2.32iT - 37T^{2} \)
41 \( 1 + 4.06T + 41T^{2} \)
43 \( 1 + 4.14iT - 43T^{2} \)
53 \( 1 + 3.40iT - 53T^{2} \)
59 \( 1 - 7.25T + 59T^{2} \)
61 \( 1 + 8.40T + 61T^{2} \)
67 \( 1 - 12.1iT - 67T^{2} \)
71 \( 1 - 3.21T + 71T^{2} \)
73 \( 1 + 2.87iT - 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 - 15.9iT - 83T^{2} \)
89 \( 1 + 6.55T + 89T^{2} \)
97 \( 1 - 2.09iT - 97T^{2} \)
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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

−10.46261359900107790941453176429, −9.902093145641637243750329279288, −8.521873842972241352533352525801, −8.136796223884603292610139401174, −7.16066181110970327867161812720, −6.19814375665637423786116564743, −5.21245486420860462436141568775, −4.60265981548442219256283531737, −3.77682299136418261185590046976, −2.62563021412960292194654653797, 0.13899908764331318779398840115, 1.79762450002978647854424281618, 2.50327176517297627110500130567, 2.85937727049917098445894072705, 4.88075028002780078950177929440, 6.03448351319758939383881902636, 6.43944639055879490811961759316, 7.51509361422142801413647823015, 8.369936882240558533885495763771, 8.840347089877969826670829176336

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