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Properties

Label 2-775-155.154-c0-0-0
Degree $2$
Conductor $775$
Sign $-0.447 - 0.894i$
Analytic cond. $0.386775$
Root an. cond. $0.621912$
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  + i·2-s + i·7-s + i·8-s − 9-s − 14-s − 16-s − i·18-s + 19-s + 31-s + i·38-s − 41-s − 2i·47-s − 56-s + 59-s + i·62-s − i·63-s + ⋯
L(s)  = 1  + i·2-s + i·7-s + i·8-s − 9-s − 14-s − 16-s − i·18-s + 19-s + 31-s + i·38-s − 41-s − 2i·47-s − 56-s + 59-s + i·62-s − i·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(0.386775\)
Root analytic conductor: \(0.621912\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (774, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :0),\ -0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.000504625\)
\(L(\frac12)\) \(\approx\) \(1.000504625\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.88094925849513100475007253196, −9.736479655856421221788304309806, −8.708947130671630556292987312140, −8.301946708134444097654891517075, −7.28272962814183108869302836062, −6.36018236574266475286208299262, −5.60804298330032465500801713255, −4.99224872250018706175975538306, −3.22838924950130307453140274912, −2.21450966206841532962125146589, 1.12751217975685847919452715463, 2.64667848140516305275747409975, 3.47417145786663930246880135241, 4.52161811020678925869205945602, 5.80229200938498819358004024042, 6.82923687497072555163317355472, 7.64333590180011217867953441274, 8.695966476342903621559896603555, 9.745375666044587925445917159751, 10.31611005555746883682549417850

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