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Properties

Label 1-1725-1725.284-r1-0-0
Degree $1$
Conductor $1725$
Sign $0.989 - 0.145i$
Analytic cond. $185.376$
Root an. cond. $185.376$
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  + (−0.0285 − 0.999i)2-s + (−0.998 + 0.0570i)4-s + (−0.415 − 0.909i)7-s + (0.0855 + 0.996i)8-s + (0.564 + 0.825i)11-s + (0.736 + 0.676i)13-s + (−0.897 + 0.441i)14-s + (0.993 − 0.113i)16-s + (−0.998 − 0.0570i)17-s + (−0.362 − 0.931i)19-s + (0.809 − 0.587i)22-s + (0.654 − 0.755i)26-s + (0.466 + 0.884i)28-s + (0.362 − 0.931i)29-s + (0.0855 + 0.996i)31-s + (−0.142 − 0.989i)32-s + ⋯
L(s)  = 1  + (−0.0285 − 0.999i)2-s + (−0.998 + 0.0570i)4-s + (−0.415 − 0.909i)7-s + (0.0855 + 0.996i)8-s + (0.564 + 0.825i)11-s + (0.736 + 0.676i)13-s + (−0.897 + 0.441i)14-s + (0.993 − 0.113i)16-s + (−0.998 − 0.0570i)17-s + (−0.362 − 0.931i)19-s + (0.809 − 0.587i)22-s + (0.654 − 0.755i)26-s + (0.466 + 0.884i)28-s + (0.362 − 0.931i)29-s + (0.0855 + 0.996i)31-s + (−0.142 − 0.989i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1725\)    =    \(3 \cdot 5^{2} \cdot 23\)
Sign: $0.989 - 0.145i$
Analytic conductor: \(185.376\)
Root analytic conductor: \(185.376\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1725} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1725,\ (1:\ ),\ 0.989 - 0.145i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.225671603 - 0.08963774001i\)
\(L(\frac12)\) \(\approx\) \(1.225671603 - 0.08963774001i\)
\(L(1)\) \(\approx\) \(0.7880550029 - 0.3911651541i\)
\(L(1)\) \(\approx\) \(0.7880550029 - 0.3911651541i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.0285 - 0.999i)T \)
7 \( 1 + (-0.415 - 0.909i)T \)
11 \( 1 + (0.564 + 0.825i)T \)
13 \( 1 + (0.736 + 0.676i)T \)
17 \( 1 + (-0.998 - 0.0570i)T \)
19 \( 1 + (-0.362 - 0.931i)T \)
29 \( 1 + (0.362 - 0.931i)T \)
31 \( 1 + (0.0855 + 0.996i)T \)
37 \( 1 + (0.985 + 0.170i)T \)
41 \( 1 + (-0.897 - 0.441i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.870 - 0.491i)T \)
59 \( 1 + (0.736 + 0.676i)T \)
61 \( 1 + (0.516 - 0.856i)T \)
67 \( 1 + (-0.610 - 0.791i)T \)
71 \( 1 + (-0.941 + 0.336i)T \)
73 \( 1 + (0.254 - 0.967i)T \)
79 \( 1 + (0.198 + 0.980i)T \)
83 \( 1 + (-0.466 + 0.884i)T \)
89 \( 1 + (0.921 + 0.389i)T \)
97 \( 1 + (0.466 + 0.884i)T \)
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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

−19.958601754083285618184967397881, −19.14159241884726760598227618919, −18.51530734992342893021427508952, −17.92834031821817984245850894141, −17.074768036242540256755288575887, −16.19594048742321733884533261559, −15.88401380788541338848019358660, −14.8958310663677327953214124081, −14.481429097956464612318134572592, −13.28104737412764801064061266950, −13.03681992074681852462898314850, −11.97276101616313475649203881181, −11.076448247959564962494577250299, −10.08572408165480961885449024634, −9.23165217060438195690807128869, −8.55197023479755450279119713513, −8.08499475728080115547406315289, −6.88336839551282177412050071855, −6.057674074604139584086332186801, −5.8113811614550876867586286994, −4.65034796612611288834845759839, −3.728156656955121062642108923507, −2.91001298383384534480240914217, −1.48938877382201983638425620830, −0.31915310363223375810455796566, 0.72898678178792352160128909917, 1.66594447531205999449914981373, 2.5528941678304677098766401318, 3.646680157362638493147374381381, 4.290504479888049928507674019373, 4.88887461548927727423846093528, 6.36298076636234850049098342255, 6.9110121846764121753369724717, 8.03611020331214463459341359694, 8.98585304808920777463953448016, 9.51151757604280773328501308089, 10.37788845165841762679939222632, 11.0567455393991428646509039105, 11.714326597548920477594612442294, 12.60654948382526593860709099252, 13.37074058762499347615902678685, 13.80759877144926032656918475364, 14.69624559795443199096258181351, 15.65100407123645067281654941811, 16.56134350998786681669373961653, 17.500939501042096372204035459900, 17.728655219426576045528923729219, 18.8710021884829572249136969732, 19.50998995194217258744177839830, 20.06106614730965098894812132562

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