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Properties

Label 2-15e2-15.8-c1-0-3
Degree $2$
Conductor $225$
Sign $-0.876 + 0.481i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 − 1.73i)2-s + 3.99i·4-s + (1.22 − 1.22i)7-s + (3.46 − 3.46i)8-s − 4.24i·11-s + (−3.67 − 3.67i)13-s − 4.24·14-s − 3.99·16-s + (1.73 + 1.73i)17-s − 5i·19-s + (−7.34 + 7.34i)22-s + (−1.73 + 1.73i)23-s + 12.7i·26-s + (4.89 + 4.89i)28-s − 4.24·29-s + ⋯
L(s)  = 1  + (−1.22 − 1.22i)2-s + 1.99i·4-s + (0.462 − 0.462i)7-s + (1.22 − 1.22i)8-s − 1.27i·11-s + (−1.01 − 1.01i)13-s − 1.13·14-s − 0.999·16-s + (0.420 + 0.420i)17-s − 1.14i·19-s + (−1.56 + 1.56i)22-s + (−0.361 + 0.361i)23-s + 2.49i·26-s + (0.925 + 0.925i)28-s − 0.787·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.876 + 0.481i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.876 + 0.481i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.142229 - 0.554215i\)
\(L(\frac12)\) \(\approx\) \(0.142229 - 0.554215i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (1.73 + 1.73i)T + 2iT^{2} \)
7 \( 1 + (-1.22 + 1.22i)T - 7iT^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
13 \( 1 + (3.67 + 3.67i)T + 13iT^{2} \)
17 \( 1 + (-1.73 - 1.73i)T + 17iT^{2} \)
19 \( 1 + 5iT - 19T^{2} \)
23 \( 1 + (1.73 - 1.73i)T - 23iT^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + (-2.44 + 2.44i)T - 37iT^{2} \)
41 \( 1 + 8.48iT - 41T^{2} \)
43 \( 1 + (1.22 + 1.22i)T + 43iT^{2} \)
47 \( 1 + (-5.19 - 5.19i)T + 47iT^{2} \)
53 \( 1 + (-6.92 + 6.92i)T - 53iT^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + (-3.67 + 3.67i)T - 67iT^{2} \)
71 \( 1 - 8.48iT - 71T^{2} \)
73 \( 1 + (-2.44 - 2.44i)T + 73iT^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 + (1.73 - 1.73i)T - 83iT^{2} \)
89 \( 1 + 8.48T + 89T^{2} \)
97 \( 1 + (8.57 - 8.57i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.47590850967784301914375074342, −10.82830197752965162340494667071, −10.06358246872476802783446505034, −9.046415633833492609780552262527, −8.137131401640558919117113142037, −7.34616539281897897653365323572, −5.48793683036491656874411270155, −3.73322246618618312885169621328, −2.51179118645682904468238432214, −0.71881077159523051674139860729, 1.91646681515026274660386773448, 4.59764290092272401539857748534, 5.73695267404867297466864619600, 6.94932558777755585588789359843, 7.62843142538659275611409059202, 8.609706231969304663220920502309, 9.663364362850317436610656346382, 10.08642708235202175042730428718, 11.62457466386676957517898233086, 12.46124525936878305766155436051

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