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Properties

Label 2-775-31.28-c1-0-45
Degree $2$
Conductor $775$
Sign $-0.990 + 0.136i$
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  + (2.12 − 1.54i)2-s + (0.110 − 1.05i)3-s + (1.51 − 4.66i)4-s + (−1.39 − 2.41i)6-s + (−4.56 + 0.969i)7-s + (−2.35 − 7.25i)8-s + (1.83 + 0.389i)9-s + (−2.65 − 2.94i)11-s + (−4.75 − 2.11i)12-s + (1.03 − 0.462i)13-s + (−8.20 + 9.10i)14-s + (−8.28 − 6.02i)16-s + (0.833 − 0.926i)17-s + (4.50 − 2.00i)18-s + (2.94 + 1.30i)19-s + ⋯
L(s)  = 1  + (1.50 − 1.09i)2-s + (0.0639 − 0.608i)3-s + (0.757 − 2.33i)4-s + (−0.568 − 0.985i)6-s + (−1.72 + 0.366i)7-s + (−0.833 − 2.56i)8-s + (0.611 + 0.129i)9-s + (−0.799 − 0.887i)11-s + (−1.37 − 0.610i)12-s + (0.288 − 0.128i)13-s + (−2.19 + 2.43i)14-s + (−2.07 − 1.50i)16-s + (0.202 − 0.224i)17-s + (1.06 − 0.472i)18-s + (0.674 + 0.300i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.990 + 0.136i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (276, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ -0.990 + 0.136i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.201823 - 2.93416i\)
\(L(\frac12)\) \(\approx\) \(0.201823 - 2.93416i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 - 5.15i)T \)
good2 \( 1 + (-2.12 + 1.54i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.110 + 1.05i)T + (-2.93 - 0.623i)T^{2} \)
7 \( 1 + (4.56 - 0.969i)T + (6.39 - 2.84i)T^{2} \)
11 \( 1 + (2.65 + 2.94i)T + (-1.14 + 10.9i)T^{2} \)
13 \( 1 + (-1.03 + 0.462i)T + (8.69 - 9.66i)T^{2} \)
17 \( 1 + (-0.833 + 0.926i)T + (-1.77 - 16.9i)T^{2} \)
19 \( 1 + (-2.94 - 1.30i)T + (12.7 + 14.1i)T^{2} \)
23 \( 1 + (1.01 + 3.12i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-7.26 + 5.28i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + (-1.58 - 2.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.357 - 3.39i)T + (-40.1 + 8.52i)T^{2} \)
43 \( 1 + (-11.3 - 5.05i)T + (28.7 + 31.9i)T^{2} \)
47 \( 1 + (3.12 + 2.26i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.0302 + 0.00642i)T + (48.4 + 21.5i)T^{2} \)
59 \( 1 + (-0.144 + 1.37i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 - 0.316T + 61T^{2} \)
67 \( 1 + (-4.42 + 7.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.69 + 1.63i)T + (64.8 + 28.8i)T^{2} \)
73 \( 1 + (-6.70 - 7.44i)T + (-7.63 + 72.6i)T^{2} \)
79 \( 1 + (8.88 - 9.86i)T + (-8.25 - 78.5i)T^{2} \)
83 \( 1 + (-0.788 - 7.50i)T + (-81.1 + 17.2i)T^{2} \)
89 \( 1 + (1.75 - 5.40i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (4.64 - 14.2i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.12888530512989263115719110962, −9.558743293569671080208569350179, −8.128829996348300476298880034592, −6.77102025370238958041412614846, −6.18232357335567140965037589030, −5.38812792169475822354535327141, −4.16745405091645713204258446083, −3.10081583110854937273194150666, −2.57940678648273601340603497138, −0.925452920449375526199855731433, 2.80699205458210805466660853587, 3.67473589026705758408680466560, 4.36554032095230872281845540676, 5.37478378221262752301415620637, 6.23503541629406882774261100059, 7.13421827066081722927337544694, 7.53223307485218141959563060286, 9.013018445306570792743832589139, 9.857828271026306100700756904976, 10.62463485923405690961962065328

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