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Properties

Label 1-33e2-1089.376-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.933 + 0.358i$
Analytic cond. $117.029$
Root an. cond. $117.029$
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  + (−0.861 − 0.508i)2-s + (0.483 + 0.875i)4-s + (−0.179 + 0.983i)5-s + (0.830 + 0.556i)7-s + (0.0285 − 0.999i)8-s + (0.654 − 0.755i)10-s + (0.905 + 0.424i)13-s + (−0.432 − 0.901i)14-s + (−0.532 + 0.846i)16-s + (0.998 − 0.0570i)17-s + (0.254 + 0.967i)19-s + (−0.948 + 0.318i)20-s + (−0.786 − 0.618i)23-s + (−0.935 − 0.353i)25-s + (−0.564 − 0.825i)26-s + ⋯
L(s)  = 1  + (−0.861 − 0.508i)2-s + (0.483 + 0.875i)4-s + (−0.179 + 0.983i)5-s + (0.830 + 0.556i)7-s + (0.0285 − 0.999i)8-s + (0.654 − 0.755i)10-s + (0.905 + 0.424i)13-s + (−0.432 − 0.901i)14-s + (−0.532 + 0.846i)16-s + (0.998 − 0.0570i)17-s + (0.254 + 0.967i)19-s + (−0.948 + 0.318i)20-s + (−0.786 − 0.618i)23-s + (−0.935 − 0.353i)25-s + (−0.564 − 0.825i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.933 + 0.358i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.933 + 0.358i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1538773313 + 0.8299137417i\)
\(L(\frac12)\) \(\approx\) \(0.1538773313 + 0.8299137417i\)
\(L(1)\) \(\approx\) \(0.7247316762 + 0.1660896865i\)
\(L(1)\) \(\approx\) \(0.7247316762 + 0.1660896865i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.861 - 0.508i)T \)
5 \( 1 + (-0.179 + 0.983i)T \)
7 \( 1 + (0.830 + 0.556i)T \)
13 \( 1 + (0.905 + 0.424i)T \)
17 \( 1 + (0.998 - 0.0570i)T \)
19 \( 1 + (0.254 + 0.967i)T \)
23 \( 1 + (-0.786 - 0.618i)T \)
29 \( 1 + (-0.161 + 0.986i)T \)
31 \( 1 + (-0.290 + 0.956i)T \)
37 \( 1 + (-0.921 - 0.389i)T \)
41 \( 1 + (-0.953 + 0.299i)T \)
43 \( 1 + (-0.723 - 0.690i)T \)
47 \( 1 + (-0.595 - 0.803i)T \)
53 \( 1 + (-0.466 + 0.884i)T \)
59 \( 1 + (-0.217 - 0.976i)T \)
61 \( 1 + (-0.00951 - 0.999i)T \)
67 \( 1 + (0.580 + 0.814i)T \)
71 \( 1 + (-0.362 + 0.931i)T \)
73 \( 1 + (0.985 - 0.170i)T \)
79 \( 1 + (-0.380 + 0.924i)T \)
83 \( 1 + (-0.272 + 0.962i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (-0.179 - 0.983i)T \)
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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

−20.70829965881608610045892848251, −20.10496333023633361256800404419, −19.364835793511698421565754947411, −18.41139077497208451273234983852, −17.64111862093173681004431889765, −17.08450860801661576220663790983, −16.335474035236309427308772503913, −15.610903947661555761295678745504, −14.88923828751792834195775242305, −13.81655538996565741813102901905, −13.256926929597694043302194507573, −11.85536392570978277854815026350, −11.38230226364846548525996815497, −10.36079081722288999646707117018, −9.59748892300591795061950484699, −8.68682273170615729236763090313, −7.97289461256466043264399704218, −7.51995674170802334505084181353, −6.23877795242144061533182767274, −5.39956337232325289667259294446, −4.64973298646012608985951583389, −3.491203123887680421253817305926, −1.86535437103633731825431119038, −1.132437072097854789191727488965, −0.245746660051939964152425621580, 1.382111614747665682862151692976, 2.04269646303985142838054626060, 3.2491030489855964719595863089, 3.80348739098783047399618906852, 5.28083297260312076265239505031, 6.36867187901553492225393791483, 7.17942538471645553700951211733, 8.14252896191383939689046105599, 8.58718270022393180325215890340, 9.768875057874908500115618213307, 10.49382152718420325182580182401, 11.16139149692400550902022536814, 11.92653084449511303735439158594, 12.49707090694606270633912055515, 13.949706024799128274436367857529, 14.45812495853065430934506608437, 15.50672142265243327772149958062, 16.20321846447943953492774079012, 17.07286138390508516172758969995, 18.12200241134541626882818103267, 18.47268260905731358446361065109, 18.92093519775923822029407618819, 20.04945932367737202547062193317, 20.731948522936097731796028737520, 21.52227986572292091643962058946

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