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Properties

Label 1-605-605.159-r0-0-0
Degree $1$
Conductor $605$
Sign $-0.101 - 0.994i$
Analytic cond. $2.80960$
Root an. cond. $2.80960$
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  + (−0.0855 + 0.996i)2-s + (−0.309 − 0.951i)3-s + (−0.985 − 0.170i)4-s + (0.974 − 0.226i)6-s + (0.564 − 0.825i)7-s + (0.254 − 0.967i)8-s + (−0.809 + 0.587i)9-s + (0.142 + 0.989i)12-s + (−0.696 − 0.717i)13-s + (0.774 + 0.633i)14-s + (0.941 + 0.336i)16-s + (0.870 − 0.491i)17-s + (−0.516 − 0.856i)18-s + (−0.736 + 0.676i)19-s + (−0.959 − 0.281i)21-s + ⋯
L(s)  = 1  + (−0.0855 + 0.996i)2-s + (−0.309 − 0.951i)3-s + (−0.985 − 0.170i)4-s + (0.974 − 0.226i)6-s + (0.564 − 0.825i)7-s + (0.254 − 0.967i)8-s + (−0.809 + 0.587i)9-s + (0.142 + 0.989i)12-s + (−0.696 − 0.717i)13-s + (0.774 + 0.633i)14-s + (0.941 + 0.336i)16-s + (0.870 − 0.491i)17-s + (−0.516 − 0.856i)18-s + (−0.736 + 0.676i)19-s + (−0.959 − 0.281i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.101 - 0.994i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ -0.101 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4970570089 - 0.5501198432i\)
\(L(\frac12)\) \(\approx\) \(0.4970570089 - 0.5501198432i\)
\(L(1)\) \(\approx\) \(0.7672158618 - 0.07880126855i\)
\(L(1)\) \(\approx\) \(0.7672158618 - 0.07880126855i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0855 + 0.996i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (0.564 - 0.825i)T \)
13 \( 1 + (-0.696 - 0.717i)T \)
17 \( 1 + (0.870 - 0.491i)T \)
19 \( 1 + (-0.736 + 0.676i)T \)
23 \( 1 + (0.959 - 0.281i)T \)
29 \( 1 + (0.993 - 0.113i)T \)
31 \( 1 + (-0.466 - 0.884i)T \)
37 \( 1 + (-0.897 - 0.441i)T \)
41 \( 1 + (-0.921 - 0.389i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 + (-0.516 + 0.856i)T \)
53 \( 1 + (-0.941 + 0.336i)T \)
59 \( 1 + (-0.921 + 0.389i)T \)
61 \( 1 + (0.0855 + 0.996i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (0.198 - 0.980i)T \)
73 \( 1 + (0.0285 - 0.999i)T \)
79 \( 1 + (-0.362 + 0.931i)T \)
83 \( 1 + (-0.610 - 0.791i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (0.998 - 0.0570i)T \)
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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

−23.18895171459166394471955593206, −21.97993390442304531468448001485, −21.56409511267933674029748532960, −21.13734907889769033560488049526, −20.0831352336016755116953665521, −19.25683477955505258517996037223, −18.45363915272432258333333358220, −17.37312294428355316021047970986, −16.9715956166600294303968820345, −15.71141894567434452924878001955, −14.7436465593299039765678196760, −14.245142033751051647173770970221, −12.862529117549720655897047101393, −11.990294872815704734268571154090, −11.41982688226548594671170419024, −10.53201630040060500255391221043, −9.72873749282313507487142633743, −8.88545490550412368884805747833, −8.24128338406762589228260985357, −6.57889072783508179097058668972, −5.088324113759950006792680269546, −4.9185766738851702451100767695, −3.609880787725947007244381181759, −2.70856923173765025607444984593, −1.51334989147056130516834564630, 0.42575960800191236549491782653, 1.59221690769753515131663096229, 3.22088012586598450723623726773, 4.644739253067838043464212267028, 5.384370119808841180763467025763, 6.400077115788027981837782860449, 7.31222077187669218654949550031, 7.81963274924512873146713648034, 8.66507322354096701610861351588, 10.015251743077482825622844046731, 10.79839594065809177889188053297, 12.10050357554245114643799664508, 12.84052401076916326907952992663, 13.75861479083191208254204296368, 14.39197157124385230849631440088, 15.195694634047594380943763394552, 16.5391866639610369326513577243, 17.04831807783030907902175822233, 17.66336135762392147310955696406, 18.562782192082205783631045748792, 19.23073029714646429947133723747, 20.22379263582979957254423394704, 21.298339896119954092451938195469, 22.58682591160731853040955911656, 23.014253748720180876218193762095

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