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Properties

Label 1-15e2-225.196-r0-0-0
Degree $1$
Conductor $225$
Sign $0.999 + 0.0279i$
Analytic cond. $1.04489$
Root an. cond. $1.04489$
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  + (−0.104 + 0.994i)2-s + (−0.978 − 0.207i)4-s + (−0.5 − 0.866i)7-s + (0.309 − 0.951i)8-s + (−0.104 + 0.994i)11-s + (−0.104 − 0.994i)13-s + (0.913 − 0.406i)14-s + (0.913 + 0.406i)16-s + (0.309 − 0.951i)17-s + (0.309 − 0.951i)19-s + (−0.978 − 0.207i)22-s + (0.913 − 0.406i)23-s + 26-s + (0.309 + 0.951i)28-s + (0.669 − 0.743i)29-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)2-s + (−0.978 − 0.207i)4-s + (−0.5 − 0.866i)7-s + (0.309 − 0.951i)8-s + (−0.104 + 0.994i)11-s + (−0.104 − 0.994i)13-s + (0.913 − 0.406i)14-s + (0.913 + 0.406i)16-s + (0.309 − 0.951i)17-s + (0.309 − 0.951i)19-s + (−0.978 − 0.207i)22-s + (0.913 − 0.406i)23-s + 26-s + (0.309 + 0.951i)28-s + (0.669 − 0.743i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.999 + 0.0279i$
Analytic conductor: \(1.04489\)
Root analytic conductor: \(1.04489\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (0:\ ),\ 0.999 + 0.0279i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8776843362 + 0.01225558160i\)
\(L(\frac12)\) \(\approx\) \(0.8776843362 + 0.01225558160i\)
\(L(1)\) \(\approx\) \(0.8434325677 + 0.1837787997i\)
\(L(1)\) \(\approx\) \(0.8434325677 + 0.1837787997i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.104 + 0.994i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (-0.104 - 0.994i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.913 - 0.406i)T \)
29 \( 1 + (0.669 - 0.743i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (0.669 + 0.743i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.669 - 0.743i)T \)
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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

−26.60011719173165502606460134293, −25.751256852305585986439460882185, −24.54167645095715262071410228574, −23.453133735185483743602994044842, −22.49675680707815245124448008400, −21.45911566500590700253063983552, −21.183146404869135742606322311454, −19.648763290075385811767160022898, −19.024524258372821050173313765240, −18.37851226160956676327612664846, −17.06428052224519914740739581842, −16.17985925713370053930106830737, −14.78003320635344927397492577761, −13.79786875181724684266386343731, −12.760675592893390969895199866266, −11.9254123637277729394146397761, −11.013815954709816227599572252315, −9.86445961564035036190413106948, −8.957499843195380767759231023471, −8.08502487095384246699921782888, −6.31190164056410015171942766446, −5.22094027233739061876406040636, −3.7794371522878066930036445014, −2.81849529970840476759143468286, −1.461055700823681773862009861628, 0.74733003715179359807946303918, 3.029653839061093913544585912742, 4.45714385688126599863530785167, 5.34962657744728325487376708194, 6.87443860030248954036183046545, 7.3080950318822704052004771529, 8.60913993761493754544118596423, 9.79465378782660962466591234374, 10.47062321399941101980635915712, 12.22625438977108239662703904117, 13.27288810421951421321488454166, 14.00120071052519378898778210872, 15.2607397126821907010940674419, 15.84442411640335024061815458159, 17.09887928061570297161435787724, 17.598739331993836880373237470480, 18.72812351669497915339762575564, 19.8209045808272081479224734162, 20.706195340519784016701910416935, 22.242064112129079990524642413629, 22.925017664920852198524391879874, 23.53577959343937664576876468779, 24.81153747540903238168274593416, 25.398776501017351982960014398357, 26.39084032154950906521504826584

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