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Properties

Label 2-1775-355.354-c0-0-2
Degree $2$
Conductor $1775$
Sign $-0.447 - 0.894i$
Analytic cond. $0.885840$
Root an. cond. $0.941190$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.445i·2-s + 1.24i·3-s + 0.801·4-s − 0.554·6-s + 0.801i·8-s − 0.554·9-s + i·12-s + 0.445·16-s − 0.246i·18-s + 0.445·19-s − 24-s + 0.554i·27-s − 1.24·29-s + i·32-s − 0.445·36-s + 0.445i·37-s + ⋯
L(s)  = 1  + 0.445i·2-s + 1.24i·3-s + 0.801·4-s − 0.554·6-s + 0.801i·8-s − 0.554·9-s + i·12-s + 0.445·16-s − 0.246i·18-s + 0.445·19-s − 24-s + 0.554i·27-s − 1.24·29-s + i·32-s − 0.445·36-s + 0.445i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1775\)    =    \(5^{2} \cdot 71\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(0.885840\)
Root analytic conductor: \(0.941190\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1775} (1774, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1775,\ (\ :0),\ -0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.415079140\)
\(L(\frac12)\) \(\approx\) \(1.415079140\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
71 \( 1 - T \)
good2 \( 1 - 0.445iT - T^{2} \)
3 \( 1 - 1.24iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 0.445T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + 1.24T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 0.445iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.80iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
73 \( 1 + 1.80iT - T^{2} \)
79 \( 1 - 1.80T + T^{2} \)
83 \( 1 + 0.445iT - T^{2} \)
89 \( 1 + 1.24T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.719236010824593967341505314148, −9.064927284139224927613547383677, −8.127099286241541708311670334560, −7.34633752673809725069611692028, −6.54811611027979861919269708460, −5.56972422622827334210074166541, −5.00329686225595433604541897385, −3.89876010504696566605932367890, −3.11590268956127141162909971875, −1.86440561665895781643089844486, 1.15730113528559608053471207462, 2.03501465456140891860281392003, 2.92925045520474225988603913289, 3.98608038445977468172567188315, 5.37207747473656920339733956356, 6.28739076740589044120160022833, 6.85316438594093364612328421251, 7.61230765170625109113244267775, 8.113358314023399852821115031391, 9.359307969187796375358476520781

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