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Properties

Label 1-605-605.424-r1-0-0
Degree $1$
Conductor $605$
Sign $-0.947 - 0.318i$
Analytic cond. $65.0162$
Root an. cond. $65.0162$
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  + (−0.998 + 0.0570i)2-s + (−0.309 − 0.951i)3-s + (0.993 − 0.113i)4-s + (0.362 + 0.931i)6-s + (−0.921 + 0.389i)7-s + (−0.985 + 0.170i)8-s + (−0.809 + 0.587i)9-s + (−0.415 − 0.909i)12-s + (−0.870 − 0.491i)13-s + (0.897 − 0.441i)14-s + (0.974 − 0.226i)16-s + (0.941 + 0.336i)17-s + (0.774 − 0.633i)18-s + (0.0285 + 0.999i)19-s + (0.654 + 0.755i)21-s + ⋯
L(s)  = 1  + (−0.998 + 0.0570i)2-s + (−0.309 − 0.951i)3-s + (0.993 − 0.113i)4-s + (0.362 + 0.931i)6-s + (−0.921 + 0.389i)7-s + (−0.985 + 0.170i)8-s + (−0.809 + 0.587i)9-s + (−0.415 − 0.909i)12-s + (−0.870 − 0.491i)13-s + (0.897 − 0.441i)14-s + (0.974 − 0.226i)16-s + (0.941 + 0.336i)17-s + (0.774 − 0.633i)18-s + (0.0285 + 0.999i)19-s + (0.654 + 0.755i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.947 - 0.318i$
Analytic conductor: \(65.0162\)
Root analytic conductor: \(65.0162\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (1:\ ),\ -0.947 - 0.318i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05327150780 - 0.3253980005i\)
\(L(\frac12)\) \(\approx\) \(0.05327150780 - 0.3253980005i\)
\(L(1)\) \(\approx\) \(0.4938573977 - 0.1282776537i\)
\(L(1)\) \(\approx\) \(0.4938573977 - 0.1282776537i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.998 + 0.0570i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (-0.921 + 0.389i)T \)
13 \( 1 + (-0.870 - 0.491i)T \)
17 \( 1 + (0.941 + 0.336i)T \)
19 \( 1 + (0.0285 + 0.999i)T \)
23 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (0.564 - 0.825i)T \)
31 \( 1 + (0.198 + 0.980i)T \)
37 \( 1 + (0.736 + 0.676i)T \)
41 \( 1 + (0.254 - 0.967i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (-0.774 - 0.633i)T \)
53 \( 1 + (-0.974 - 0.226i)T \)
59 \( 1 + (-0.254 - 0.967i)T \)
61 \( 1 + (0.998 + 0.0570i)T \)
67 \( 1 + (-0.841 - 0.540i)T \)
71 \( 1 + (0.610 + 0.791i)T \)
73 \( 1 + (0.516 + 0.856i)T \)
79 \( 1 + (-0.696 - 0.717i)T \)
83 \( 1 + (0.0855 + 0.996i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (0.466 + 0.884i)T \)
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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

−23.24947380806623074525656566187, −22.2045824022971203194115398300, −21.50863887627832430555093527082, −20.69518950221111608294607106047, −19.75419532234625466426847955390, −19.31699062362678807153767680576, −18.1565527154371765906193367743, −17.18992141250693416658022044824, −16.67392469962529085028896401165, −15.97874042994970909822200482059, −15.18566387615062744247327137432, −14.27231552889211979925661356092, −12.89010864484129268266710798630, −11.87483717098124287839397627772, −11.141385511498708067500445511009, −10.190524498140714731484472210294, −9.54952199683331681531782602651, −9.03192868180006312856214306230, −7.65023923481487465013553484665, −6.81766645379377625189942187700, −5.84561654274649243474331436977, −4.69404438663302229420510084677, −3.40582587453394917809899611452, −2.65775156847882172606627892988, −0.89793385685709544434231281667, 0.15609332059394279183323552626, 1.21519876082818867869938925910, 2.43395327472623356440337691533, 3.2158845306105193910311386938, 5.26024373897280135856069906046, 6.16757062586535847459814721860, 6.83374510096868211487333704957, 7.842896556059738165329556381925, 8.464005801601134541104835015671, 9.715266462858372324969992897801, 10.3094388178382686624234666985, 11.47191186831715749227328322550, 12.41334455462691726899385185948, 12.70398416410303132893095440310, 14.18322749992048149928564505107, 15.045217552001523826469128129, 16.177544387338979517302005589517, 16.81670256671561028171979531128, 17.55221157767325813743322774318, 18.47933062949596688285216375396, 19.093129307840162680846406561239, 19.59745421832728003867215915959, 20.56223445157688906913566645449, 21.6347770946277158945248270159, 22.72547380199803279465531685746

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