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Properties

Label 1-42e2-1764.1523-r0-0-0
Degree $1$
Conductor $1764$
Sign $-0.971 + 0.236i$
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  + (−0.0747 − 0.997i)5-s + (0.365 + 0.930i)11-s + (0.365 + 0.930i)13-s + (−0.955 + 0.294i)17-s + (0.5 − 0.866i)19-s + (−0.733 + 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.955 + 0.294i)29-s − 31-s + (−0.733 − 0.680i)37-s + (−0.826 + 0.563i)41-s + (−0.826 − 0.563i)43-s + (0.623 − 0.781i)47-s + (0.733 − 0.680i)53-s + (0.900 − 0.433i)55-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)5-s + (0.365 + 0.930i)11-s + (0.365 + 0.930i)13-s + (−0.955 + 0.294i)17-s + (0.5 − 0.866i)19-s + (−0.733 + 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.955 + 0.294i)29-s − 31-s + (−0.733 − 0.680i)37-s + (−0.826 + 0.563i)41-s + (−0.826 − 0.563i)43-s + (0.623 − 0.781i)47-s + (0.733 − 0.680i)53-s + (0.900 − 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.971 + 0.236i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ -0.971 + 0.236i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001299067065 + 0.01083521231i\)
\(L(\frac12)\) \(\approx\) \(0.001299067065 + 0.01083521231i\)
\(L(1)\) \(\approx\) \(0.8133920149 - 0.06256501526i\)
\(L(1)\) \(\approx\) \(0.8133920149 - 0.06256501526i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.0747 - 0.997i)T \)
11 \( 1 + (0.365 + 0.930i)T \)
13 \( 1 + (0.365 + 0.930i)T \)
17 \( 1 + (-0.955 + 0.294i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.733 + 0.680i)T \)
29 \( 1 + (-0.955 + 0.294i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.733 - 0.680i)T \)
41 \( 1 + (-0.826 + 0.563i)T \)
43 \( 1 + (-0.826 - 0.563i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (0.733 - 0.680i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (0.365 - 0.930i)T \)
79 \( 1 - T \)
83 \( 1 + (0.365 - 0.930i)T \)
89 \( 1 + (0.988 - 0.149i)T \)
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

−20.08661824645539624022535008055, −18.8773207894781101809479413073, −18.58153210499130881270438186608, −17.83377219659530527851888260481, −16.9815798958638311397222066600, −16.11965611430361191828216497480, −15.446144544573411272778817650318, −14.70704542429612926675028937121, −13.940479129627704120659035851, −13.398640335948584630489176428402, −12.35110792110757746816678079366, −11.5152710866992594221472166516, −10.85862668067616479143696393309, −10.26786140235605415325679425707, −9.32155284779484373357834797710, −8.38404624241804434279197696283, −7.70748957350724718925684496975, −6.77566273586198770518595156073, −6.062714178938689572702091239625, −5.36917342549784619150570468288, −4.017024843467344475195437633611, −3.40727279743578874421450788154, −2.57721809731790295431222451480, −1.488238758778980996874364611451, −0.003621744634773968904600075, 1.59188780976260592365916080410, 1.98683546347335036357013468803, 3.56275386022023828571235734384, 4.2685868718548130178865924349, 4.99596114521054908358862984011, 5.8605371163331157275247986202, 6.92718047644163234917021258412, 7.49457943292148719684681102250, 8.73751302174886786342404936956, 9.064613750404447925969673066977, 9.84302314335048270265566476673, 10.913092153674744451663515101686, 11.77953479737423214901611431646, 12.22944679277305618391245262171, 13.35173932940583251787185364946, 13.55852646169587472876086213592, 14.8042968552033841091516148661, 15.41964810908857630624021237888, 16.25164752872344596821426023557, 16.82358649930813726444604557202, 17.66119066171666245936185911874, 18.20065686687355389414397549023, 19.297080154251599018466613617041, 20.07575984785846395716466146284, 20.284550370931382532550989786495

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