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Properties

Label 1-42e2-1764.1355-r0-0-0
Degree $1$
Conductor $1764$
Sign $0.690 - 0.723i$
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  + (0.900 − 0.433i)5-s + (0.623 + 0.781i)11-s + (0.365 − 0.930i)13-s + (0.733 − 0.680i)17-s + (0.5 − 0.866i)19-s + (−0.222 + 0.974i)23-s + (0.623 − 0.781i)25-s + (−0.955 − 0.294i)29-s + (0.5 − 0.866i)31-s + (0.955 + 0.294i)37-s + (−0.826 − 0.563i)41-s + (−0.826 + 0.563i)43-s + (0.365 − 0.930i)47-s + (−0.955 + 0.294i)53-s + (0.900 + 0.433i)55-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)5-s + (0.623 + 0.781i)11-s + (0.365 − 0.930i)13-s + (0.733 − 0.680i)17-s + (0.5 − 0.866i)19-s + (−0.222 + 0.974i)23-s + (0.623 − 0.781i)25-s + (−0.955 − 0.294i)29-s + (0.5 − 0.866i)31-s + (0.955 + 0.294i)37-s + (−0.826 − 0.563i)41-s + (−0.826 + 0.563i)43-s + (0.365 − 0.930i)47-s + (−0.955 + 0.294i)53-s + (0.900 + 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.690 - 0.723i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ 0.690 - 0.723i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.942859111 - 0.8312364128i\)
\(L(\frac12)\) \(\approx\) \(1.942859111 - 0.8312364128i\)
\(L(1)\) \(\approx\) \(1.342942869 - 0.2237885254i\)
\(L(1)\) \(\approx\) \(1.342942869 - 0.2237885254i\)

Euler product

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

−20.53739123167820929916716086097, −19.43482251420601580180032841888, −18.72883627420820483246278065207, −18.33078785028851052373260637745, −17.30155082638729135858093612918, −16.629326034094603104997217918646, −16.19689826250754585443926249679, −14.85706954704873900051679207053, −14.353285372888662042265924704698, −13.82024513381970602304472279689, −12.96409277399979837515226815292, −12.074150442251134464326976296, −11.292823483230174335377646362843, −10.49658906126205094562188079373, −9.80569506308744861198224237119, −9.00376586855174485201138503048, −8.28069325849498670175339220315, −7.22276240023436608792640343571, −6.24856102456827273978472883508, −5.98496911806774029293389072899, −4.87018834185107848505684250237, −3.75912296715795313254212627360, −3.096110030485492211884496087862, −1.888790873888067390783874587737, −1.23779841714943365372158761032, 0.829850718192267413225387942138, 1.71913781182989589231831261729, 2.69812944748126687686598299951, 3.65781699913622011528106300265, 4.76571342610812489403678024633, 5.43359953832461548535046780065, 6.17188015688441295660087921671, 7.152102840145440347400485995002, 7.90942385383936526662483582489, 8.91742941746428177373619258612, 9.718285286773393085141316465514, 9.99443511198333919157946940315, 11.24953595861600627621358269725, 11.892494024970001342268164021630, 12.840717840824071306262141081239, 13.413840693901381616025161993874, 14.07557584961597848790477269684, 15.04344120018498265003489635599, 15.61374512813202461107355086506, 16.68113994378610109386088524119, 17.179269019380757665848547774509, 17.93723987803527532204208573515, 18.44912394119729834943777056042, 19.55392383433539280200714762873, 20.398612395262043796158288359922

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