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Properties

Label 2-1775-355.354-c0-0-0
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  + 1.24i·2-s + 1.80i·3-s − 0.554·4-s − 2.24·6-s + 0.554i·8-s − 2.24·9-s − 0.999i·12-s − 1.24·16-s − 2.80i·18-s − 1.24·19-s − 1.00·24-s − 2.24i·27-s + 1.80·29-s − 0.999i·32-s + 1.24·36-s + 1.24i·37-s + ⋯
L(s)  = 1  + 1.24i·2-s + 1.80i·3-s − 0.554·4-s − 2.24·6-s + 0.554i·8-s − 2.24·9-s − 0.999i·12-s − 1.24·16-s − 2.80i·18-s − 1.24·19-s − 1.00·24-s − 2.24i·27-s + 1.80·29-s − 0.999i·32-s + 1.24·36-s + 1.24i·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

\(L(\frac{1}{2})\) \(\approx\) \(0.9884947961\)
\(L(\frac12)\) \(\approx\) \(0.9884947961\)
\(L(1)\) 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 \( 1 - 1.24iT - T^{2} \)
3 \( 1 - 1.80iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.24T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - 1.80T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.24iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 0.445iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
73 \( 1 - 0.445iT - T^{2} \)
79 \( 1 - 0.445T + T^{2} \)
83 \( 1 + 1.24iT - T^{2} \)
89 \( 1 - 1.80T + T^{2} \)
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

−9.973569246872389145974470473300, −9.011624599736031633744652241444, −8.514343665656410841466517348705, −7.82674836753351024795526404246, −6.51583385579984567940682432378, −6.11514972890366561354731970582, −4.88233592532976972619071666430, −4.74859253280895333906653521796, −3.59385613048877252925626153633, −2.51451247240672084907852180966, 0.72464795215486256310658779656, 1.86342784499651700667081382145, 2.46674271335840595905999054450, 3.43841427727628657865687777637, 4.66749382274865192799432860213, 6.01194605351290792804179982173, 6.62264245016531400859719138435, 7.29865066864092663347844229721, 8.246664186995982160597830341719, 8.840270737086092308779050486215

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