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Properties

Label 2-1075-5.4-c1-0-10
Degree $2$
Conductor $1075$
Sign $-0.447 - 0.894i$
Analytic cond. $8.58391$
Root an. cond. $2.92983$
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  + 2.22i·2-s − 2.75i·3-s − 2.94·4-s + 6.11·6-s + 1.28i·7-s − 2.08i·8-s − 4.57·9-s + 0.693·11-s + 8.09i·12-s + 4.64i·13-s − 2.85·14-s − 1.23·16-s − 1.40i·17-s − 10.1i·18-s + 3.40·19-s + ⋯
L(s)  = 1  + 1.57i·2-s − 1.58i·3-s − 1.47·4-s + 2.49·6-s + 0.484i·7-s − 0.738i·8-s − 1.52·9-s + 0.209·11-s + 2.33i·12-s + 1.28i·13-s − 0.761·14-s − 0.308·16-s − 0.341i·17-s − 2.39i·18-s + 0.782·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1075\)    =    \(5^{2} \cdot 43\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(8.58391\)
Root analytic conductor: \(2.92983\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1075} (474, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1075,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.276957365\)
\(L(\frac12)\) \(\approx\) \(1.276957365\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
43 \( 1 - iT \)
good2 \( 1 - 2.22iT - 2T^{2} \)
3 \( 1 + 2.75iT - 3T^{2} \)
7 \( 1 - 1.28iT - 7T^{2} \)
11 \( 1 - 0.693T + 11T^{2} \)
13 \( 1 - 4.64iT - 13T^{2} \)
17 \( 1 + 1.40iT - 17T^{2} \)
19 \( 1 - 3.40T + 19T^{2} \)
23 \( 1 - 3.45iT - 23T^{2} \)
29 \( 1 - 1.86T + 29T^{2} \)
31 \( 1 + 4.95T + 31T^{2} \)
37 \( 1 - 11.0iT - 37T^{2} \)
41 \( 1 - 7.25T + 41T^{2} \)
47 \( 1 - 13.1iT - 47T^{2} \)
53 \( 1 - 6.42iT - 53T^{2} \)
59 \( 1 + 0.115T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 - 9.51iT - 67T^{2} \)
71 \( 1 + 0.745T + 71T^{2} \)
73 \( 1 + 4.37iT - 73T^{2} \)
79 \( 1 + 17.3T + 79T^{2} \)
83 \( 1 - 10.8iT - 83T^{2} \)
89 \( 1 + 1.62T + 89T^{2} \)
97 \( 1 + 8.69iT - 97T^{2} \)
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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

−9.615469039332536772406486395156, −8.936202898839496793860618912725, −8.176461822399670594265168214258, −7.36802793197956628352469190456, −6.95840563893210253276861206454, −6.15350278473162142129572874618, −5.53244587353711156677731175889, −4.38553236032962417402832125327, −2.66978898195202485018462988431, −1.37531385273089020552565174252, 0.61796378754525042527185623509, 2.38987565249820019533658056038, 3.51507902037133581354864155169, 3.88057052501216918803268572022, 4.90293803811081926227004171792, 5.69488788578594772570773889920, 7.26151999265308915643189081728, 8.520669614634224929911047219002, 9.198690691515015145799995249487, 10.01916946144827929312101494137

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