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Properties

Label 2-775-155.134-c1-0-24
Degree $2$
Conductor $775$
Sign $0.582 + 0.812i$
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  + (0.744 − 1.02i)2-s + (−1.47 + 0.155i)3-s + (0.122 + 0.376i)4-s + (−0.940 + 1.62i)6-s + (0.455 − 2.14i)7-s + (2.88 + 0.937i)8-s + (−0.779 + 0.165i)9-s + (−0.636 + 0.706i)11-s + (−0.238 − 0.536i)12-s + (0.0683 − 0.153i)13-s + (−1.85 − 2.06i)14-s + (2.46 − 1.79i)16-s + (4.88 − 4.40i)17-s + (−0.410 + 0.921i)18-s + (1.05 − 0.468i)19-s + ⋯
L(s)  = 1  + (0.526 − 0.724i)2-s + (−0.852 + 0.0895i)3-s + (0.0611 + 0.188i)4-s + (−0.383 + 0.664i)6-s + (0.172 − 0.809i)7-s + (1.02 + 0.331i)8-s + (−0.259 + 0.0552i)9-s + (−0.191 + 0.212i)11-s + (−0.0689 − 0.154i)12-s + (0.0189 − 0.0426i)13-s + (−0.495 − 0.550i)14-s + (0.617 − 0.448i)16-s + (1.18 − 1.06i)17-s + (−0.0967 + 0.217i)18-s + (0.241 − 0.107i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.582 + 0.812i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ 0.582 + 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49353 - 0.766977i\)
\(L(\frac12)\) \(\approx\) \(1.49353 - 0.766977i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.56 + 0.217i)T \)
good2 \( 1 + (-0.744 + 1.02i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (1.47 - 0.155i)T + (2.93 - 0.623i)T^{2} \)
7 \( 1 + (-0.455 + 2.14i)T + (-6.39 - 2.84i)T^{2} \)
11 \( 1 + (0.636 - 0.706i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (-0.0683 + 0.153i)T + (-8.69 - 9.66i)T^{2} \)
17 \( 1 + (-4.88 + 4.40i)T + (1.77 - 16.9i)T^{2} \)
19 \( 1 + (-1.05 + 0.468i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (-4.40 - 1.43i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-1.08 - 0.785i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (-3.35 - 1.93i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0343 - 0.326i)T + (-40.1 - 8.52i)T^{2} \)
43 \( 1 + (3.91 + 8.79i)T + (-28.7 + 31.9i)T^{2} \)
47 \( 1 + (3.31 + 4.56i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.52 + 7.17i)T + (-48.4 + 21.5i)T^{2} \)
59 \( 1 + (0.277 + 2.63i)T + (-57.7 + 12.2i)T^{2} \)
61 \( 1 + 1.74T + 61T^{2} \)
67 \( 1 + (-0.478 + 0.276i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.11 - 0.236i)T + (64.8 - 28.8i)T^{2} \)
73 \( 1 + (-5.88 - 5.30i)T + (7.63 + 72.6i)T^{2} \)
79 \( 1 + (3.03 + 3.37i)T + (-8.25 + 78.5i)T^{2} \)
83 \( 1 + (-0.324 - 0.0341i)T + (81.1 + 17.2i)T^{2} \)
89 \( 1 + (-4.54 - 13.9i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (14.7 - 4.79i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.47446900887164843271939006303, −9.744092397553729595897854817019, −8.349077852615364577225053381759, −7.49758313691570128906110057894, −6.71368997392701099440247535933, −5.30637157703310092596671100953, −4.83877790346926537264173878338, −3.63471614616550773930223693636, −2.68359934707531259660468758308, −0.996682963935956098935425902032, 1.24427902904957998543013224770, 2.96615071937332112841130854055, 4.50061106185914294445408423012, 5.38263446396817212731063398020, 5.94449141662925114420929280062, 6.54972755755707153596939500890, 7.72119485404718598586626118787, 8.516211985612409641489714389096, 9.699054488521138877188471482015, 10.59009250910405784947906333275

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