Properties

Label 2-15e2-225.4-c3-0-65
Degree $2$
Conductor $225$
Sign $-0.903 + 0.428i$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.922 − 4.34i)2-s + (4.70 − 2.20i)3-s + (−10.6 + 4.75i)4-s + (1.62 + 11.0i)5-s + (−13.9 − 18.3i)6-s + (8.42 − 4.86i)7-s + (9.65 + 13.2i)8-s + (17.2 − 20.7i)9-s + (46.5 − 17.2i)10-s + (36.6 − 7.79i)11-s + (−39.7 + 46.0i)12-s + (4.83 − 22.7i)13-s + (−28.8 − 32.0i)14-s + (32.0 + 48.4i)15-s + (−13.8 + 15.3i)16-s + (−74.2 − 102. i)17-s + ⋯
L(s)  = 1  + (−0.326 − 1.53i)2-s + (0.905 − 0.425i)3-s + (−1.33 + 0.594i)4-s + (0.144 + 0.989i)5-s + (−0.948 − 1.25i)6-s + (0.454 − 0.262i)7-s + (0.426 + 0.587i)8-s + (0.638 − 0.769i)9-s + (1.47 − 0.545i)10-s + (1.00 − 0.213i)11-s + (−0.956 + 1.10i)12-s + (0.103 − 0.485i)13-s + (−0.551 − 0.612i)14-s + (0.551 + 0.833i)15-s + (−0.216 + 0.240i)16-s + (−1.05 − 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.903 + 0.428i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.903 + 0.428i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.452698 - 2.00860i\)
\(L(\frac12)\) \(\approx\) \(0.452698 - 2.00860i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.70 + 2.20i)T \)
5 \( 1 + (-1.62 - 11.0i)T \)
good2 \( 1 + (0.922 + 4.34i)T + (-7.30 + 3.25i)T^{2} \)
7 \( 1 + (-8.42 + 4.86i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-36.6 + 7.79i)T + (1.21e3 - 541. i)T^{2} \)
13 \( 1 + (-4.83 + 22.7i)T + (-2.00e3 - 893. i)T^{2} \)
17 \( 1 + (74.2 + 102. i)T + (-1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (-6.89 + 5.01i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 + (-129. + 116. i)T + (1.27e3 - 1.21e4i)T^{2} \)
29 \( 1 + (20.1 - 191. i)T + (-2.38e4 - 5.07e3i)T^{2} \)
31 \( 1 + (23.2 + 221. i)T + (-2.91e4 + 6.19e3i)T^{2} \)
37 \( 1 + (165. - 53.6i)T + (4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (-315. - 67.0i)T + (6.29e4 + 2.80e4i)T^{2} \)
43 \( 1 + (-260. + 150. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-169. - 17.8i)T + (1.01e5 + 2.15e4i)T^{2} \)
53 \( 1 + (134. - 184. i)T + (-4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (-135. - 28.7i)T + (1.87e5 + 8.35e4i)T^{2} \)
61 \( 1 + (791. - 168. i)T + (2.07e5 - 9.23e4i)T^{2} \)
67 \( 1 + (397. - 41.7i)T + (2.94e5 - 6.25e4i)T^{2} \)
71 \( 1 + (-719. - 522. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (-229. - 74.5i)T + (3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (-13.4 + 127. i)T + (-4.82e5 - 1.02e5i)T^{2} \)
83 \( 1 + (-289. + 650. i)T + (-3.82e5 - 4.24e5i)T^{2} \)
89 \( 1 + (98.0 - 301. i)T + (-5.70e5 - 4.14e5i)T^{2} \)
97 \( 1 + (733. + 77.0i)T + (8.92e5 + 1.89e5i)T^{2} \)
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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

−11.20018398074928322424363369409, −10.65625586365727829881081001938, −9.383975615230337101666597972136, −8.927755176840450904186428041579, −7.48595320066560200840809698540, −6.55926144439836674084592294305, −4.31986096346400465746961287561, −3.14861603543453128000815891144, −2.34159945927989925687031684199, −0.932483710918386144350160803252, 1.71986132506659139949114172493, 4.05974740530550229458890043831, 4.96865356223782279783598167235, 6.18887526222644907446061322618, 7.37942365962873779065250211358, 8.390152381230312945177576266414, 8.964576439363482281651066665691, 9.530310100866775155832761329312, 11.11512680816530302927324669582, 12.50499403885478672197792209044

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