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Properties

Label 1-485-485.109-r0-0-0
Degree $1$
Conductor $485$
Sign $-0.936 + 0.351i$
Analytic cond. $2.25233$
Root an. cond. $2.25233$
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  + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − i·4-s − 6-s + (0.923 + 0.382i)7-s + (0.707 + 0.707i)8-s + i·9-s + (−0.707 − 0.707i)11-s + (0.707 − 0.707i)12-s + (−0.923 + 0.382i)13-s + (−0.923 + 0.382i)14-s − 16-s + (−0.923 + 0.382i)17-s + (−0.707 − 0.707i)18-s + (−0.923 + 0.382i)19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − i·4-s − 6-s + (0.923 + 0.382i)7-s + (0.707 + 0.707i)8-s + i·9-s + (−0.707 − 0.707i)11-s + (0.707 − 0.707i)12-s + (−0.923 + 0.382i)13-s + (−0.923 + 0.382i)14-s − 16-s + (−0.923 + 0.382i)17-s + (−0.707 − 0.707i)18-s + (−0.923 + 0.382i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(485\)    =    \(5 \cdot 97\)
Sign: $-0.936 + 0.351i$
Analytic conductor: \(2.25233\)
Root analytic conductor: \(2.25233\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{485} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 485,\ (0:\ ),\ -0.936 + 0.351i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1695620909 + 0.9333730111i\)
\(L(\frac12)\) \(\approx\) \(0.1695620909 + 0.9333730111i\)
\(L(1)\) \(\approx\) \(0.6666298418 + 0.5650173823i\)
\(L(1)\) \(\approx\) \(0.6666298418 + 0.5650173823i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
97 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (0.923 + 0.382i)T \)
11 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 + (-0.923 + 0.382i)T \)
19 \( 1 + (-0.923 + 0.382i)T \)
23 \( 1 + (0.382 + 0.923i)T \)
29 \( 1 + (-0.382 - 0.923i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
43 \( 1 + iT \)
47 \( 1 + iT \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (-0.382 + 0.923i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.923 + 0.382i)T \)
71 \( 1 + (0.382 - 0.923i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + (0.923 - 0.382i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.49011541340041192017006849987, −22.41324434322483280593527564237, −21.30524739929523282412373955117, −20.50391927919552157630238080670, −20.09913964027944093022781172013, −19.15550447431679146142357901386, −18.302112367644040760230635542877, −17.63159249437253993520094542699, −17.03026953951107920721329043039, −15.51702345634204658419622760706, −14.705128818421244260401482017, −13.62472380654741789139084913505, −12.79493995639531842765857676087, −12.13721000813861042886642771572, −10.98719656962537429202511440470, −10.22926698173428199922691462569, −9.063937651030856810310190118118, −8.36520359974325277610100093621, −7.42630311269885908512389997662, −6.91789566104511449405768395575, −4.947387888605163112662349161018, −3.95209428537722159660648999659, −2.42841851219947298735247742918, −2.13163760014226615423346790691, −0.57051343526476794838428019332, 1.7342785897311249388296486626, 2.69415785384453103209449855596, 4.39938592664688775609171940756, 5.07520740430957735193338060815, 6.183270755307974097237879226081, 7.56512815932039696699855447993, 8.2418736056399863064937595109, 8.91764383705948053362456302846, 9.87486987054133159704727948291, 10.7558048508061952468706372032, 11.52058084095558172546536494896, 13.23373936542431791323373449096, 14.08686311371002031958703970078, 14.98263870110692384877755261048, 15.37068844229796687397008128855, 16.3985988466908566699926135051, 17.21348237842301483526502590698, 18.065043717880918185316214998015, 19.22637578569122693693616586739, 19.51826779461711387945535279455, 20.86397704182771684712169615382, 21.35081990331379803865309533653, 22.37666839232600256600770680083, 23.63600087898521888234276927948, 24.37343401570505023157539225503

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