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Properties

Label 2-1075-5.4-c1-0-34
Degree $2$
Conductor $1075$
Sign $0.894 - 0.447i$
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  + 2i·2-s − 2i·3-s − 2·4-s + 4·6-s − 9-s + 3·11-s + 4i·12-s − 5i·13-s − 4·16-s + 3i·17-s − 2i·18-s + 2·19-s + 6i·22-s − i·23-s + 10·26-s − 4i·27-s + ⋯
L(s)  = 1  + 1.41i·2-s − 1.15i·3-s − 4-s + 1.63·6-s − 0.333·9-s + 0.904·11-s + 1.15i·12-s − 1.38i·13-s − 16-s + 0.727i·17-s − 0.471i·18-s + 0.458·19-s + 1.27i·22-s − 0.208i·23-s + 1.96·26-s − 0.769i·27-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1075\)    =    \(5^{2} \cdot 43\)
Sign: $0.894 - 0.447i$
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: \((2,\ 1075,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.724304523\)
\(L(\frac12)\) \(\approx\) \(1.724304523\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 2T^{2} \)
3 \( 1 + 2iT - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 + 5iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 3iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 15iT - 83T^{2} \)
89 \( 1 - 4T + 89T^{2} \)
97 \( 1 + 7iT - 97T^{2} \)
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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

−9.710887369035426443261044105689, −8.546681828173521378243024331044, −8.129965403794831182849467838140, −7.29589947142409344562829979625, −6.70099659312056784552669654279, −5.99468633262745676277317437391, −5.21773867052717109715799487418, −3.93179156198282917234609400869, −2.42428671521233822914668873517, −0.953298339373886933243585073199, 1.23062939970639946144566310459, 2.52108619379382811414155029462, 3.63854218768032536755905089511, 4.23583843722116955224224300924, 4.97679095399392265340890894007, 6.41295841238528194521222493379, 7.29369786750731142335093871617, 8.885487520408565285330406713451, 9.287604040986603424513083812796, 9.887892592088091902631424159776

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