Properties

Label 2-775-5.4-c1-0-31
Degree $2$
Conductor $775$
Sign $-0.447 + 0.894i$
Analytic cond. $6.18840$
Root an. cond. $2.48765$
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  i·2-s − 2i·3-s + 4-s − 2·6-s + 4i·7-s − 3i·8-s − 9-s + 4·11-s − 2i·12-s + 4·14-s − 16-s − 8i·17-s + i·18-s − 4·19-s + 8·21-s − 4i·22-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.15i·3-s + 0.5·4-s − 0.816·6-s + 1.51i·7-s − 1.06i·8-s − 0.333·9-s + 1.20·11-s − 0.577i·12-s + 1.06·14-s − 0.250·16-s − 1.94i·17-s + 0.235i·18-s − 0.917·19-s + 1.74·21-s − 0.852i·22-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02139 - 1.65265i\)
\(L(\frac12)\) \(\approx\) \(1.02139 - 1.65265i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 - T \)
good2 \( 1 + iT - 2T^{2} \)
3 \( 1 + 2iT - 3T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 8iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 2iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 2iT - 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + 18iT - 97T^{2} \)
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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

−10.01830143414690191502730580649, −9.209691558422406042896636052929, −8.414203308227145010359011096816, −7.22419766342925323737336405580, −6.61320130409679507781861861624, −5.89712295289510896925541882761, −4.46428612386640225674599382818, −2.87952974221328543037634165706, −2.25967921686028348629757301002, −1.09708263069770751014201669031, 1.63695749679435195545684141020, 3.63719687322004884698017978240, 4.09824571581183166026255145169, 5.18641750815580665502027089007, 6.51502813927159837972874893201, 6.81206925472201560951082659878, 8.074202638186350387474567878816, 8.720113529259264515940044608652, 10.03966517541177843272733279293, 10.40531401436142140076783551920

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