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Properties

Label 2-1775-355.354-c0-0-1
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  + 1.80i·2-s − 0.445i·3-s − 2.24·4-s + 0.801·6-s − 2.24i·8-s + 0.801·9-s + i·12-s + 1.80·16-s + 1.44i·18-s + 1.80·19-s − 1.00·24-s − 0.801i·27-s + 0.445·29-s + 1.00i·32-s − 1.80·36-s + 1.80i·37-s + ⋯
L(s)  = 1  + 1.80i·2-s − 0.445i·3-s − 2.24·4-s + 0.801·6-s − 2.24i·8-s + 0.801·9-s + i·12-s + 1.80·16-s + 1.44i·18-s + 1.80·19-s − 1.00·24-s − 0.801i·27-s + 0.445·29-s + 1.00i·32-s − 1.80·36-s + 1.80i·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

\(L(\frac{1}{2})\) \(\approx\) \(1.079808613\)
\(L(\frac12)\) \(\approx\) \(1.079808613\)
\(L(1)\) 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 \( 1 - 1.80iT - T^{2} \)
3 \( 1 + 0.445iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 1.80T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - 0.445T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.80iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.24iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
73 \( 1 - 1.24iT - T^{2} \)
79 \( 1 + 1.24T + T^{2} \)
83 \( 1 + 1.80iT - T^{2} \)
89 \( 1 - 0.445T + T^{2} \)
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

−9.703821838208654933130308519392, −8.555588187499920793042921765636, −7.934329799209165703371234247783, −7.27963019542155882575776146665, −6.70070401317642684164834843689, −5.94668584793679278077128790293, −5.04341852054521978197543415389, −4.39267957355157035999478368087, −3.15519378475634476919851249189, −1.26125751219645572983642261369, 1.07706411477148863910027749614, 2.18137248751873857462139645112, 3.29211515206819184090920198774, 3.90583587036341142541527303477, 4.81217205050003746082539900201, 5.51479169028511106931578650809, 6.98034889422931010104277437681, 7.88409088864041493502283669895, 8.969169186019108832355666034288, 9.516722043609338237858983198042

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