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Properties

Label 1-42e2-1764.1499-r0-0-0
Degree $1$
Conductor $1764$
Sign $0.281 + 0.959i$
Analytic cond. $8.19198$
Root an. cond. $8.19198$
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  + (−0.826 − 0.563i)5-s + (−0.988 + 0.149i)11-s + (0.365 − 0.930i)13-s + (0.222 + 0.974i)17-s − 19-s + (0.955 − 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.955 − 0.294i)29-s + (0.5 + 0.866i)31-s + (−0.222 − 0.974i)37-s + (−0.826 − 0.563i)41-s + (−0.826 + 0.563i)43-s + (−0.988 + 0.149i)47-s + (0.222 − 0.974i)53-s + (0.900 + 0.433i)55-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)5-s + (−0.988 + 0.149i)11-s + (0.365 − 0.930i)13-s + (0.222 + 0.974i)17-s − 19-s + (0.955 − 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.955 − 0.294i)29-s + (0.5 + 0.866i)31-s + (−0.222 − 0.974i)37-s + (−0.826 − 0.563i)41-s + (−0.826 + 0.563i)43-s + (−0.988 + 0.149i)47-s + (0.222 − 0.974i)53-s + (0.900 + 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.281 + 0.959i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ 0.281 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5464247670 + 0.4093246144i\)
\(L(\frac12)\) \(\approx\) \(0.5464247670 + 0.4093246144i\)
\(L(1)\) \(\approx\) \(0.7736077355 + 0.01769409111i\)
\(L(1)\) \(\approx\) \(0.7736077355 + 0.01769409111i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.826 - 0.563i)T \)
11 \( 1 + (-0.988 + 0.149i)T \)
13 \( 1 + (0.365 - 0.930i)T \)
17 \( 1 + (0.222 + 0.974i)T \)
19 \( 1 - T \)
23 \( 1 + (0.955 - 0.294i)T \)
29 \( 1 + (-0.955 - 0.294i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.222 - 0.974i)T \)
41 \( 1 + (-0.826 - 0.563i)T \)
43 \( 1 + (-0.826 + 0.563i)T \)
47 \( 1 + (-0.988 + 0.149i)T \)
53 \( 1 + (0.222 - 0.974i)T \)
59 \( 1 + (0.0747 + 0.997i)T \)
61 \( 1 + (0.955 + 0.294i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.365 + 0.930i)T \)
89 \( 1 + (-0.623 + 0.781i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−20.14009949084861804663471935078, −19.06449470381511231668417668439, −18.7562572300602251514504758687, −18.17352349614287918423882493977, −16.94179369659860010309679871473, −16.45764199715729290215713944842, −15.44039245580826562638915320100, −15.15834310326262202807855928140, −14.14777706360634514674739976218, −13.414917691068977463955194414342, −12.65167160178798208538456982826, −11.56293779520692182455199508559, −11.2797560243625885835346525927, −10.38623881086962947090962459898, −9.53767675817507626377101559833, −8.54549979421066676025022044120, −7.904161420837669305912919624228, −7.02023090305918352283144979134, −6.46223993668500443397135575914, −5.24814394440367332430806676126, −4.519840060904852317399664448850, −3.53259968799939369286561094141, −2.825527820079091626626496264974, −1.79768664434058026902192699343, −0.29418784814802863698925133218, 0.925803034248372158951691191106, 2.10645790132210577434126932423, 3.23225217507064512633716791517, 3.93500070274514258683614035128, 4.950897092956718783641560861482, 5.530402415219016954855388235096, 6.64192719611718305348220987668, 7.55627592601420133594000659904, 8.33514866604460316573689242302, 8.67999004060549486244537685878, 9.96602781106461404489637341671, 10.6765050871692167719095307569, 11.29268661988287648384086211880, 12.397725409664881775198306578149, 12.86606180514526901696300861677, 13.402488607674055899654041539049, 14.87262092368831281606661687481, 15.08926409291530311614476966202, 15.968424997604729622645876022232, 16.62514108477326720558316898594, 17.43772459608442708357604299802, 18.168528856135887689887660545474, 19.17848017763646557137688765104, 19.464368974086526442040869399718, 20.58569805004832222123163945829

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