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Properties

Label 2-15e2-9.7-c1-0-14
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: $\chi_{225} (151, \cdot )$
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

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

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