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Properties

Label 1-605-605.404-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  + (−0.254 − 0.967i)2-s + (0.809 − 0.587i)3-s + (−0.870 + 0.491i)4-s + (−0.774 − 0.633i)6-s + (0.974 − 0.226i)7-s + (0.696 + 0.717i)8-s + (0.309 − 0.951i)9-s + (−0.415 + 0.909i)12-s + (−0.736 − 0.676i)13-s + (−0.466 − 0.884i)14-s + (0.516 − 0.856i)16-s + (−0.0285 − 0.999i)17-s + (−0.998 − 0.0570i)18-s + (−0.610 + 0.791i)19-s + (0.654 − 0.755i)21-s + ⋯
L(s)  = 1  + (−0.254 − 0.967i)2-s + (0.809 − 0.587i)3-s + (−0.870 + 0.491i)4-s + (−0.774 − 0.633i)6-s + (0.974 − 0.226i)7-s + (0.696 + 0.717i)8-s + (0.309 − 0.951i)9-s + (−0.415 + 0.909i)12-s + (−0.736 − 0.676i)13-s + (−0.466 − 0.884i)14-s + (0.516 − 0.856i)16-s + (−0.0285 − 0.999i)17-s + (−0.998 − 0.0570i)18-s + (−0.610 + 0.791i)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: $\chi_{605} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (1:\ ),\ -0.947 + 0.318i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2963602428 - 1.810255321i\)
\(L(\frac12)\) \(\approx\) \(-0.2963602428 - 1.810255321i\)
\(L(1)\) \(\approx\) \(0.7900154815 - 0.8562032533i\)
\(L(1)\) \(\approx\) \(0.7900154815 - 0.8562032533i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.254 - 0.967i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (0.974 - 0.226i)T \)
13 \( 1 + (-0.736 - 0.676i)T \)
17 \( 1 + (-0.0285 - 0.999i)T \)
19 \( 1 + (-0.610 + 0.791i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (-0.941 - 0.336i)T \)
31 \( 1 + (0.993 - 0.113i)T \)
37 \( 1 + (-0.198 + 0.980i)T \)
41 \( 1 + (0.362 - 0.931i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (0.998 - 0.0570i)T \)
53 \( 1 + (-0.516 - 0.856i)T \)
59 \( 1 + (-0.362 - 0.931i)T \)
61 \( 1 + (0.254 - 0.967i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
71 \( 1 + (-0.564 - 0.825i)T \)
73 \( 1 + (0.0855 + 0.996i)T \)
79 \( 1 + (-0.897 - 0.441i)T \)
83 \( 1 + (-0.921 - 0.389i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (0.985 + 0.170i)T \)
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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

−23.63021972075933095373916127113, −22.45571955601900274523538372399, −21.60754423682868265283648076161, −21.07091940786103463628587045392, −19.79584208247046496168109318470, −19.200814918547706844708426682206, −18.30470539604237893364464050562, −17.25269696315296670555996538227, −16.70426148569251226521429771686, −15.62632082767533966723775216543, −14.82157566049538038933862260268, −14.55505781174599212282201399987, −13.55623729235604755470164724445, −12.60480052967576022524200274864, −11.10127698177797262436869584604, −10.33563216584085284793585804519, −9.21514268213666452034993448390, −8.70333877502984523899487080575, −7.84532913432221541361084449975, −7.02443877173212563605833924048, −5.772306751165032923797442985536, −4.623505473701182240331093934187, −4.272061545580807485892381486054, −2.62501414618265882385071056149, −1.452095223894345395004354485034, 0.43009423510456507516683509222, 1.53076135758813903457117097356, 2.389432627272019846807392140212, 3.35264145796691437589146251523, 4.40696236748449346231463515150, 5.44271478217653133902107650986, 7.16988105165367190639901603054, 7.84059088422584732501501577574, 8.595987362514388084747279537924, 9.542339551444475100419416981360, 10.37651860718960671835675711254, 11.480622501268039351912669588453, 12.16678592948284268451199117861, 13.076292601746647400006501831, 13.85763431031138572465966252443, 14.55031350466598475837362590557, 15.46408686511756205967614168231, 17.15832006873668206601989098417, 17.49639771914503278795272129643, 18.615061033588455108351179413901, 19.017930433300017941199620842069, 20.0929624724687223213326189062, 20.62524701089803787502513709483, 21.18213039395346280214910764401, 22.299429607283642127167094542646

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