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Properties

Label 2-15e2-9.4-c1-0-6
Degree $2$
Conductor $225$
Sign $-0.435 - 0.900i$
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  + (1.31 + 2.28i)2-s + (1.71 − 0.238i)3-s + (−2.46 + 4.27i)4-s + (2.80 + 3.59i)6-s + (−0.898 − 1.55i)7-s − 7.73·8-s + (2.88 − 0.817i)9-s + (−0.904 − 1.56i)11-s + (−3.21 + 7.92i)12-s + (0.985 − 1.70i)13-s + (2.36 − 4.09i)14-s + (−5.24 − 9.08i)16-s − 4.80·17-s + (5.66 + 5.50i)18-s + 2.96·19-s + ⋯
L(s)  = 1  + (0.931 + 1.61i)2-s + (0.990 − 0.137i)3-s + (−1.23 + 2.13i)4-s + (1.14 + 1.46i)6-s + (−0.339 − 0.588i)7-s − 2.73·8-s + (0.962 − 0.272i)9-s + (−0.272 − 0.472i)11-s + (−0.928 + 2.28i)12-s + (0.273 − 0.473i)13-s + (0.632 − 1.09i)14-s + (−1.31 − 2.27i)16-s − 1.16·17-s + (1.33 + 1.29i)18-s + 0.680·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.435 - 0.900i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.435 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19608 + 1.90739i\)
\(L(\frac12)\) \(\approx\) \(1.19608 + 1.90739i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 + 0.238i)T \)
5 \( 1 \)
good2 \( 1 + (-1.31 - 2.28i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.898 + 1.55i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.904 + 1.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.985 + 1.70i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.80T + 17T^{2} \)
19 \( 1 - 2.96T + 19T^{2} \)
23 \( 1 + (0.866 - 1.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.68 - 6.38i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.31 + 2.27i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.6T + 37T^{2} \)
41 \( 1 + (-1.23 + 2.13i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.63 + 6.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.14 - 5.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.72T + 53T^{2} \)
59 \( 1 + (5.51 - 9.54i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.33 - 10.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.55 + 7.88i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.27T + 71T^{2} \)
73 \( 1 - 3.58T + 73T^{2} \)
79 \( 1 + (1.05 + 1.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.549 - 0.951i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + (1.91 + 3.31i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.14505681402237271819929707803, −12.19411204340790790743640889575, −10.47058921438187700836827712845, −9.037971771767855210882769183938, −8.344266282892643734840606376631, −7.32591617061031348968389003846, −6.67795214498774458615507671693, −5.35903575050058841501206157076, −4.09196684861710781850445062800, −3.14227492041090969235174519077, 1.93906875327518460728216172118, 2.91380833931535232614665378200, 4.07651293462280429439623591632, 5.05705682128586080715690144014, 6.61433964381561562279763689970, 8.432385332764309875613366581789, 9.401016282956138286765030848214, 10.05660354281747267040182979821, 11.09949058915450714640623313885, 12.11127864702231656412401718791

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