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Properties

Label 2-755-5.4-c1-0-56
Degree $2$
Conductor $755$
Sign $0.270 + 0.962i$
Analytic cond. $6.02870$
Root an. cond. $2.45534$
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  + 2.50i·2-s − 2.26i·3-s − 4.26·4-s + (−0.605 − 2.15i)5-s + 5.66·6-s + 2.41i·7-s − 5.66i·8-s − 2.12·9-s + (5.38 − 1.51i)10-s + 0.723·11-s + 9.64i·12-s + 1.06i·13-s − 6.05·14-s + (−4.87 + 1.36i)15-s + 5.64·16-s − 6.18i·17-s + ⋯
L(s)  = 1  + 1.76i·2-s − 1.30i·3-s − 2.13·4-s + (−0.270 − 0.962i)5-s + 2.31·6-s + 0.913i·7-s − 2.00i·8-s − 0.706·9-s + (1.70 − 0.478i)10-s + 0.218·11-s + 2.78i·12-s + 0.294i·13-s − 1.61·14-s + (−1.25 + 0.353i)15-s + 1.41·16-s − 1.49i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(755\)    =    \(5 \cdot 151\)
Sign: $0.270 + 0.962i$
Analytic conductor: \(6.02870\)
Root analytic conductor: \(2.45534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{755} (454, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 755,\ (\ :1/2),\ 0.270 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.477070 - 0.361422i\)
\(L(\frac12)\) \(\approx\) \(0.477070 - 0.361422i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.605 + 2.15i)T \)
151 \( 1 + T \)
good2 \( 1 - 2.50iT - 2T^{2} \)
3 \( 1 + 2.26iT - 3T^{2} \)
7 \( 1 - 2.41iT - 7T^{2} \)
11 \( 1 - 0.723T + 11T^{2} \)
13 \( 1 - 1.06iT - 13T^{2} \)
17 \( 1 + 6.18iT - 17T^{2} \)
19 \( 1 + 5.28T + 19T^{2} \)
23 \( 1 + 3.16iT - 23T^{2} \)
29 \( 1 + 6.98T + 29T^{2} \)
31 \( 1 + 7.00T + 31T^{2} \)
37 \( 1 + 7.55iT - 37T^{2} \)
41 \( 1 - 0.455T + 41T^{2} \)
43 \( 1 - 4.49iT - 43T^{2} \)
47 \( 1 + 0.268iT - 47T^{2} \)
53 \( 1 + 5.33iT - 53T^{2} \)
59 \( 1 + 1.64T + 59T^{2} \)
61 \( 1 - 6.49T + 61T^{2} \)
67 \( 1 + 9.19iT - 67T^{2} \)
71 \( 1 + 7.00T + 71T^{2} \)
73 \( 1 + 1.54iT - 73T^{2} \)
79 \( 1 + 5.23T + 79T^{2} \)
83 \( 1 - 12.4iT - 83T^{2} \)
89 \( 1 - 0.0292T + 89T^{2} \)
97 \( 1 - 12.3iT - 97T^{2} \)
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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

−9.402760840182152891677653096162, −8.969789130382387845436617846629, −8.202278455268723126309349125797, −7.46194833816589273783363366228, −6.80196748114882400862981821393, −5.90119968964083613684745046973, −5.20900695395018940279007429671, −4.16947483889149017265925645242, −2.08292348899942520862028536158, −0.30124267363188914652765381598, 1.81267689852022301794358311742, 3.25218499310496269266614851945, 3.88877573623264722088621441606, 4.33452373496664101206846092953, 5.73777665635372734852145126953, 7.16365447770749174481003787990, 8.395940687643648560604148544466, 9.317756677412616892050971761030, 10.16583781921296151724428151573, 10.56602508013114240755241589687

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