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Properties

Label 1-42e2-1764.1235-r1-0-0
Degree $1$
Conductor $1764$
Sign $0.00356 + 0.999i$
Analytic cond. $189.568$
Root an. cond. $189.568$
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  + (0.365 − 0.930i)5-s + (0.955 + 0.294i)11-s + (−0.955 − 0.294i)13-s + (0.0747 + 0.997i)17-s + (−0.5 + 0.866i)19-s + (0.826 + 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.0747 − 0.997i)29-s + 31-s + (0.826 − 0.563i)37-s + (−0.988 + 0.149i)41-s + (0.988 + 0.149i)43-s + (0.222 + 0.974i)47-s + (−0.826 − 0.563i)53-s + (0.623 − 0.781i)55-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)5-s + (0.955 + 0.294i)11-s + (−0.955 − 0.294i)13-s + (0.0747 + 0.997i)17-s + (−0.5 + 0.866i)19-s + (0.826 + 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.0747 − 0.997i)29-s + 31-s + (0.826 − 0.563i)37-s + (−0.988 + 0.149i)41-s + (0.988 + 0.149i)43-s + (0.222 + 0.974i)47-s + (−0.826 − 0.563i)53-s + (0.623 − 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00356 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00356 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.00356 + 0.999i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ 0.00356 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9797184250 + 0.9762349696i\)
\(L(\frac12)\) \(\approx\) \(0.9797184250 + 0.9762349696i\)
\(L(1)\) \(\approx\) \(1.065665714 + 0.02359426298i\)
\(L(1)\) \(\approx\) \(1.065665714 + 0.02359426298i\)

Euler product

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

−19.79260841948371030394752808802, −19.01723873166640599313069702005, −18.5736834896288394883912671266, −17.5224646058848200902712632462, −17.11892148850332622951484194408, −16.24822367289169675376689918874, −15.23407375119917074045772603882, −14.65119829152529608926795284895, −14.030112436988307736719047503120, −13.33070776545549725242236590711, −12.28686572435631741665873921363, −11.54367779409813496783506478197, −10.910243086065702023105837724572, −10.02202893418190443548401611841, −9.32605952274270013837158182741, −8.6129636876910537477012706568, −7.32715673760620533373963591194, −6.87999487138841631013156279184, −6.17819463942496651275424072877, −5.06728027735503504540286146556, −4.32403490137236019379478784732, −3.089288729645652339984283152164, −2.612424524859594139440803284936, −1.472715093922575605088366947673, −0.25537829021754258388569694523, 1.03948797413292692055414730115, 1.74673357772191789172620323903, 2.79990284119892263957430370588, 4.0894211920762978697427929953, 4.53867850019957396699428548221, 5.66602203460420679572874899757, 6.198822128336665681084298755466, 7.313367041567797019992130311765, 8.13331728732732002756299385797, 8.88545636263776652913488220834, 9.69903124552797440278344358031, 10.20451571711570922122323667942, 11.37706937500606207570035726199, 12.16784537300615109619119779219, 12.71188660035576918660545799109, 13.424566727659994460437056696495, 14.42181604626166482970832537489, 14.945695991535922750180924718651, 15.85370661481684591884341345414, 16.82206637102080697225076959111, 17.22810362083919290625597305527, 17.69385429493632325299655492959, 19.099503417572027938690152459865, 19.4089670721615433731083158005, 20.26353006061533515688233762222

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