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Properties

Label 1-675-675.2-r0-0-0
Degree $1$
Conductor $675$
Sign $-0.960 + 0.277i$
Analytic cond. $3.13468$
Root an. cond. $3.13468$
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  + (0.788 + 0.615i)2-s + (0.241 + 0.970i)4-s + (0.642 + 0.766i)7-s + (−0.406 + 0.913i)8-s + (−0.990 − 0.139i)11-s + (−0.788 + 0.615i)13-s + (0.0348 + 0.999i)14-s + (−0.882 + 0.469i)16-s + (−0.994 + 0.104i)17-s + (−0.913 − 0.406i)19-s + (−0.694 − 0.719i)22-s + (0.529 + 0.848i)23-s − 26-s + (−0.587 + 0.809i)28-s + (0.559 + 0.829i)29-s + ⋯
L(s)  = 1  + (0.788 + 0.615i)2-s + (0.241 + 0.970i)4-s + (0.642 + 0.766i)7-s + (−0.406 + 0.913i)8-s + (−0.990 − 0.139i)11-s + (−0.788 + 0.615i)13-s + (0.0348 + 0.999i)14-s + (−0.882 + 0.469i)16-s + (−0.994 + 0.104i)17-s + (−0.913 − 0.406i)19-s + (−0.694 − 0.719i)22-s + (0.529 + 0.848i)23-s − 26-s + (−0.587 + 0.809i)28-s + (0.559 + 0.829i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.960 + 0.277i$
Analytic conductor: \(3.13468\)
Root analytic conductor: \(3.13468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ -0.960 + 0.277i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2233673849 + 1.576033469i\)
\(L(\frac12)\) \(\approx\) \(0.2233673849 + 1.576033469i\)
\(L(1)\) \(\approx\) \(1.066039149 + 0.8667300047i\)
\(L(1)\) \(\approx\) \(1.066039149 + 0.8667300047i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.788 + 0.615i)T \)
7 \( 1 + (0.642 + 0.766i)T \)
11 \( 1 + (-0.990 - 0.139i)T \)
13 \( 1 + (-0.788 + 0.615i)T \)
17 \( 1 + (-0.994 + 0.104i)T \)
19 \( 1 + (-0.913 - 0.406i)T \)
23 \( 1 + (0.529 + 0.848i)T \)
29 \( 1 + (0.559 + 0.829i)T \)
31 \( 1 + (-0.997 - 0.0697i)T \)
37 \( 1 + (0.207 - 0.978i)T \)
41 \( 1 + (0.615 + 0.788i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (0.0697 + 0.997i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (0.990 - 0.139i)T \)
61 \( 1 + (-0.374 - 0.927i)T \)
67 \( 1 + (0.829 + 0.559i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (0.207 + 0.978i)T \)
79 \( 1 + (-0.559 - 0.829i)T \)
83 \( 1 + (0.275 + 0.961i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (0.898 + 0.438i)T \)
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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

−22.45365438422045167976511557502, −21.44381949638003628885364302370, −20.782233327995132929069667107060, −20.156378140146564046594468938001, −19.366111546973627271582229616162, −18.37997331068143248795305981049, −17.57154432781299500986926482217, −16.58355344976689147047394108645, −15.410327265202801688923096436590, −14.87199833584084232486074904331, −13.96027366038793977271932649005, −13.12857867643714980132888588816, −12.523115061417299398586706293030, −11.441481204164430918570994355532, −10.5498004690916054504999161302, −10.225566133385423048285279624712, −8.84272381046189369801771562106, −7.705639215759802723589931451, −6.82271725343721104947709634446, −5.659224989790816919890860284985, −4.73167183130605459557332146522, −4.13384898931141673012592568522, −2.76813905595469657906904005545, −2.02806971535532664865228422929, −0.53964683944804399253895144612, 2.0830985928655245113627801813, 2.734004122165637927913696962291, 4.15273089009746658953557932383, 4.97494205433752401850719055094, 5.6682715933538776478122365625, 6.782948596792409480089460662767, 7.60143747440324419212663437530, 8.546640613987252163301123834912, 9.27778385744377562312527940889, 10.89490043432299489863522492958, 11.42901878242690688644934946162, 12.59517247754731750491458472927, 13.033602745552452266603832335, 14.206580651516084135396842276305, 14.82226433858726790716342268974, 15.60836632155489763301187349910, 16.27934142363702325332320968344, 17.44832150340858883760167355149, 17.88076540410112983655606952928, 19.01132648677778679928970225006, 20.00359713689641022240337298813, 21.121760546773128550460505668510, 21.553032894748671461178681478154, 22.182420320029134681627317842342, 23.34318925593600552428129102823

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