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Properties

Label 2-15e2-9.7-c3-0-0
Degree $2$
Conductor $225$
Sign $0.399 + 0.916i$
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  + (−1.68 + 2.92i)2-s + (1.5 + 4.97i)3-s + (−1.68 − 2.92i)4-s + (−17.0 − 4.00i)6-s + (0.813 − 1.40i)7-s − 15.6·8-s + (−22.5 + 14.9i)9-s + (16.4 − 28.4i)11-s + (12 − 12.7i)12-s + (−16.5 − 28.5i)13-s + (2.74 + 4.75i)14-s + (39.8 − 68.9i)16-s − 110.·17-s + (−5.64 − 90.8i)18-s − 54.3·19-s + ⋯
L(s)  = 1  + (−0.596 + 1.03i)2-s + (0.288 + 0.957i)3-s + (−0.210 − 0.365i)4-s + (−1.16 − 0.272i)6-s + (0.0439 − 0.0761i)7-s − 0.689·8-s + (−0.833 + 0.552i)9-s + (0.450 − 0.780i)11-s + (0.288 − 0.307i)12-s + (−0.352 − 0.610i)13-s + (0.0523 + 0.0907i)14-s + (0.621 − 1.07i)16-s − 1.57·17-s + (−0.0739 − 1.18i)18-s − 0.655·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.399 + 0.916i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.399 + 0.916i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0910408 - 0.0596239i\)
\(L(\frac12)\) \(\approx\) \(0.0910408 - 0.0596239i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 4.97i)T \)
5 \( 1 \)
good2 \( 1 + (1.68 - 2.92i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-0.813 + 1.40i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-16.4 + 28.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (16.5 + 28.5i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 110.T + 4.91e3T^{2} \)
19 \( 1 + 54.3T + 6.85e3T^{2} \)
23 \( 1 + (33.7 + 58.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (137. - 237. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 347.T + 5.06e4T^{2} \)
41 \( 1 + (145. + 252. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-100. + 174. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (241. - 417. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 175.T + 1.48e5T^{2} \)
59 \( 1 + (-91.6 - 158. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (218. - 378. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (415. + 720. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 118.T + 3.57e5T^{2} \)
73 \( 1 + 183.T + 3.89e5T^{2} \)
79 \( 1 + (-319. + 552. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (747. - 1.29e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.43e3T + 7.04e5T^{2} \)
97 \( 1 + (445. - 772. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.61574335431421086344041237966, −11.28557181378304750879958581562, −10.52468348911904739158477718427, −9.208637009584605109268003893142, −8.763898289642580835858149830652, −7.78914781165290771645070955015, −6.56694936494609542919417219350, −5.55578797917333274848042374668, −4.18785495279735504264892554070, −2.79330741214283807549331319971, 0.04909348965796730914412840664, 1.72468330353516380338735233105, 2.47372598937004086693397737244, 4.16338526774365846917216523538, 6.08905854487404015798892050070, 6.98490219050015745506896519217, 8.262761809034376118415677436515, 9.194794582840104861441804280766, 9.915998766807955508512554262641, 11.40756628378383529206161081869

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