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Properties

Label 2-15e2-225.34-c1-0-5
Degree $2$
Conductor $225$
Sign $0.841 + 0.540i$
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  + (−0.359 − 0.806i)2-s + (−1.53 + 0.803i)3-s + (0.816 − 0.907i)4-s + (−2.07 + 0.822i)5-s + (1.19 + 0.948i)6-s + (4.01 + 2.31i)7-s + (−2.70 − 0.878i)8-s + (1.70 − 2.46i)9-s + (1.41 + 1.38i)10-s + (4.61 − 2.05i)11-s + (−0.524 + 2.04i)12-s + (0.606 − 1.36i)13-s + (0.427 − 4.06i)14-s + (2.52 − 2.93i)15-s + (0.00718 + 0.0683i)16-s + (0.802 + 0.260i)17-s + ⋯
L(s)  = 1  + (−0.253 − 0.570i)2-s + (−0.885 + 0.463i)3-s + (0.408 − 0.453i)4-s + (−0.929 + 0.367i)5-s + (0.489 + 0.387i)6-s + (1.51 + 0.875i)7-s + (−0.956 − 0.310i)8-s + (0.569 − 0.821i)9-s + (0.445 + 0.436i)10-s + (1.39 − 0.619i)11-s + (−0.151 + 0.591i)12-s + (0.168 − 0.377i)13-s + (0.114 − 1.08i)14-s + (0.653 − 0.757i)15-s + (0.00179 + 0.0170i)16-s + (0.194 + 0.0632i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.841 + 0.540i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.841 + 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.914821 - 0.268696i\)
\(L(\frac12)\) \(\approx\) \(0.914821 - 0.268696i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.53 - 0.803i)T \)
5 \( 1 + (2.07 - 0.822i)T \)
good2 \( 1 + (0.359 + 0.806i)T + (-1.33 + 1.48i)T^{2} \)
7 \( 1 + (-4.01 - 2.31i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.61 + 2.05i)T + (7.36 - 8.17i)T^{2} \)
13 \( 1 + (-0.606 + 1.36i)T + (-8.69 - 9.66i)T^{2} \)
17 \( 1 + (-0.802 - 0.260i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.0226 + 0.0696i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-6.49 - 0.683i)T + (22.4 + 4.78i)T^{2} \)
29 \( 1 + (-0.298 - 0.0633i)T + (26.4 + 11.7i)T^{2} \)
31 \( 1 + (4.35 - 0.926i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (-3.58 - 4.92i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.14 - 1.84i)T + (27.4 + 30.4i)T^{2} \)
43 \( 1 + (10.7 + 6.21i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.126 + 0.596i)T + (-42.9 - 19.1i)T^{2} \)
53 \( 1 + (9.68 - 3.14i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (4.55 + 2.02i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (-4.96 + 2.21i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (1.48 + 6.98i)T + (-61.2 + 27.2i)T^{2} \)
71 \( 1 + (-1.16 - 3.59i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.52 - 10.3i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.136 - 0.0291i)T + (72.1 + 32.1i)T^{2} \)
83 \( 1 + (-6.44 + 5.80i)T + (8.67 - 82.5i)T^{2} \)
89 \( 1 + (2.12 + 1.54i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.80 + 8.51i)T + (-88.6 - 39.4i)T^{2} \)
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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

−11.62977702049559733830485522776, −11.33818922301403521288362796554, −10.70163536258859495728501739681, −9.353965122662659286282813594623, −8.432894901559873201715875780552, −6.95469377089555423290058902196, −5.88310186895482656183903808044, −4.79109909346338766507190394128, −3.33583256596758855165890722627, −1.29004687598902248632704337724, 1.41027507844478139034714005075, 4.02135320082362920067903292737, 4.98175126119998607638837750406, 6.59719763990532691565282899515, 7.34595351368460335339075336946, 7.960442831933393607725539813695, 9.088432723821281818566764709578, 10.92676366347242809654671588210, 11.44336868631493520738119291539, 12.04411000240381877791980503966

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