Properties

Label 2-15e2-225.106-c1-0-9
Degree $2$
Conductor $225$
Sign $0.909 + 0.416i$
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  + (−1.39 − 0.296i)2-s + (−0.399 + 1.68i)3-s + (0.0344 + 0.0153i)4-s + (−2.10 − 0.756i)5-s + (1.05 − 2.23i)6-s + (1.20 − 2.08i)7-s + (2.26 + 1.64i)8-s + (−2.68 − 1.34i)9-s + (2.71 + 1.68i)10-s + (0.0309 + 0.00656i)11-s + (−0.0395 + 0.0518i)12-s + (6.17 − 1.31i)13-s + (−2.30 + 2.55i)14-s + (2.11 − 3.24i)15-s + (−2.72 − 3.02i)16-s + (2.72 + 1.97i)17-s + ⋯
L(s)  = 1  + (−0.987 − 0.209i)2-s + (−0.230 + 0.973i)3-s + (0.0172 + 0.00765i)4-s + (−0.941 − 0.338i)5-s + (0.432 − 0.912i)6-s + (0.455 − 0.788i)7-s + (0.801 + 0.582i)8-s + (−0.893 − 0.449i)9-s + (0.858 + 0.531i)10-s + (0.00931 + 0.00198i)11-s + (−0.0114 + 0.0149i)12-s + (1.71 − 0.363i)13-s + (−0.615 + 0.683i)14-s + (0.546 − 0.837i)15-s + (−0.681 − 0.756i)16-s + (0.659 + 0.479i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.909 + 0.416i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.909 + 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.557291 - 0.121551i\)
\(L(\frac12)\) \(\approx\) \(0.557291 - 0.121551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.399 - 1.68i)T \)
5 \( 1 + (2.10 + 0.756i)T \)
good2 \( 1 + (1.39 + 0.296i)T + (1.82 + 0.813i)T^{2} \)
7 \( 1 + (-1.20 + 2.08i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.0309 - 0.00656i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-6.17 + 1.31i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (-2.72 - 1.97i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.81 - 1.31i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.28 + 1.42i)T + (-2.40 - 22.8i)T^{2} \)
29 \( 1 + (0.977 + 9.29i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (-0.393 + 3.73i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-3.22 + 9.94i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.58 - 0.337i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (5.58 - 9.67i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.330 - 3.14i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (-1.23 + 0.896i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-12.5 + 2.66i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (10.3 + 2.20i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (-0.237 + 2.26i)T + (-65.5 - 13.9i)T^{2} \)
71 \( 1 + (3.63 - 2.64i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.380 - 1.17i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.0961 + 0.915i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (-5.07 + 2.26i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (0.932 + 2.86i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.08 - 10.3i)T + (-94.8 + 20.1i)T^{2} \)
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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

−11.56260174414187491515907115053, −11.05034297881298980827365749381, −10.28540848653170903589856289516, −9.302150686673749004943744727573, −8.293071551176849952870366082438, −7.77442574102206289771993658480, −5.86593442422373215870300941591, −4.50817977403360005720950256512, −3.69739657683625024996186145039, −0.868550875409427512505221783306, 1.27503814768059339124837710296, 3.38060194594487674637653900661, 5.16387513688833444801684930428, 6.65155098038879207514372809907, 7.44398289877251206049053279103, 8.503912054757202818151429300532, 8.792269363810061245128645468808, 10.48408613398839698776297096659, 11.41867307236631866694289705979, 12.04592133887050028244335411474

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