Properties

Label 2-15e2-225.121-c1-0-21
Degree $2$
Conductor $225$
Sign $-0.997 - 0.0705i$
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  + (−1.16 + 0.247i)2-s + (−1.18 − 1.26i)3-s + (−0.528 + 0.235i)4-s + (0.918 − 2.03i)5-s + (1.69 + 1.18i)6-s + (0.157 + 0.272i)7-s + (2.48 − 1.80i)8-s + (−0.205 + 2.99i)9-s + (−0.566 + 2.60i)10-s + (−2.82 + 0.601i)11-s + (0.922 + 0.390i)12-s + (−6.38 − 1.35i)13-s + (−0.251 − 0.279i)14-s + (−3.66 + 1.24i)15-s + (−1.67 + 1.86i)16-s + (0.794 − 0.577i)17-s + ⋯
L(s)  = 1  + (−0.824 + 0.175i)2-s + (−0.682 − 0.730i)3-s + (−0.264 + 0.117i)4-s + (0.410 − 0.911i)5-s + (0.690 + 0.483i)6-s + (0.0595 + 0.103i)7-s + (0.879 − 0.638i)8-s + (−0.0684 + 0.997i)9-s + (−0.179 + 0.823i)10-s + (−0.852 + 0.181i)11-s + (0.266 + 0.112i)12-s + (−1.77 − 0.376i)13-s + (−0.0671 − 0.0746i)14-s + (−0.946 + 0.321i)15-s + (−0.419 + 0.465i)16-s + (0.192 − 0.139i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.997 - 0.0705i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.997 - 0.0705i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00508833 + 0.144076i\)
\(L(\frac12)\) \(\approx\) \(0.00508833 + 0.144076i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.18 + 1.26i)T \)
5 \( 1 + (-0.918 + 2.03i)T \)
good2 \( 1 + (1.16 - 0.247i)T + (1.82 - 0.813i)T^{2} \)
7 \( 1 + (-0.157 - 0.272i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.82 - 0.601i)T + (10.0 - 4.47i)T^{2} \)
13 \( 1 + (6.38 + 1.35i)T + (11.8 + 5.28i)T^{2} \)
17 \( 1 + (-0.794 + 0.577i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (5.88 - 4.27i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.876 + 0.973i)T + (-2.40 + 22.8i)T^{2} \)
29 \( 1 + (-0.450 + 4.28i)T + (-28.3 - 6.02i)T^{2} \)
31 \( 1 + (-0.905 - 8.61i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (-0.00738 - 0.0227i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (4.48 + 0.952i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (1.34 + 2.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.544 + 5.18i)T + (-45.9 - 9.77i)T^{2} \)
53 \( 1 + (3.44 + 2.50i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (5.06 + 1.07i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 + (-7.24 + 1.54i)T + (55.7 - 24.8i)T^{2} \)
67 \( 1 + (-0.260 - 2.47i)T + (-65.5 + 13.9i)T^{2} \)
71 \( 1 + (-9.63 - 7.00i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.283 - 0.872i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.18 + 11.3i)T + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (9.08 + 4.04i)T + (55.5 + 61.6i)T^{2} \)
89 \( 1 + (-3.78 + 11.6i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-0.772 + 7.35i)T + (-94.8 - 20.1i)T^{2} \)
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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

−12.07277478507881444456343022215, −10.36196249813299097837311843232, −9.970117364121616836923195225996, −8.545160082568407436903044332979, −7.938022785437791264706778753271, −6.90623346048856639806053040945, −5.43890291424914945353224510886, −4.64444001049620372191596039973, −2.01969669324218830824315205462, −0.16146367930413292197339113739, 2.49544543280569964478458192081, 4.43006779824891678622376180493, 5.40070015451744110461547494779, 6.72691459234444329588848127499, 7.86742546822741529620595375317, 9.286092416978092162590088625873, 9.876770545305431243944123391728, 10.67300888370035578390609617817, 11.23280728503617656609062236079, 12.56020897013585000619761597895

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