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Properties

Label 2-15e2-225.31-c1-0-26
Degree $2$
Conductor $225$
Sign $-0.736 + 0.676i$
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  + (1.53 − 1.70i)2-s + (−0.365 − 1.69i)3-s + (−0.343 − 3.26i)4-s + (0.322 + 2.21i)5-s + (−3.45 − 1.98i)6-s + (−1.62 − 2.80i)7-s + (−2.39 − 1.73i)8-s + (−2.73 + 1.23i)9-s + (4.27 + 2.85i)10-s + (1.30 − 1.45i)11-s + (−5.40 + 1.77i)12-s + (2.38 + 2.64i)13-s + (−7.28 − 1.54i)14-s + (3.62 − 1.35i)15-s + (−0.216 + 0.0460i)16-s + (0.726 + 0.527i)17-s + ⋯
L(s)  = 1  + (1.08 − 1.20i)2-s + (−0.210 − 0.977i)3-s + (−0.171 − 1.63i)4-s + (0.144 + 0.989i)5-s + (−1.41 − 0.808i)6-s + (−0.612 − 1.06i)7-s + (−0.845 − 0.613i)8-s + (−0.910 + 0.412i)9-s + (1.35 + 0.902i)10-s + (0.393 − 0.437i)11-s + (−1.56 + 0.512i)12-s + (0.661 + 0.734i)13-s + (−1.94 − 0.414i)14-s + (0.936 − 0.349i)15-s + (−0.0541 + 0.0115i)16-s + (0.176 + 0.128i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.736 + 0.676i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.736 + 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.687977 - 1.76722i\)
\(L(\frac12)\) \(\approx\) \(0.687977 - 1.76722i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.365 + 1.69i)T \)
5 \( 1 + (-0.322 - 2.21i)T \)
good2 \( 1 + (-1.53 + 1.70i)T + (-0.209 - 1.98i)T^{2} \)
7 \( 1 + (1.62 + 2.80i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.30 + 1.45i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (-2.38 - 2.64i)T + (-1.35 + 12.9i)T^{2} \)
17 \( 1 + (-0.726 - 0.527i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.733 - 0.532i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-8.31 - 1.76i)T + (21.0 + 9.35i)T^{2} \)
29 \( 1 + (6.82 + 3.03i)T + (19.4 + 21.5i)T^{2} \)
31 \( 1 + (7.07 - 3.15i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (1.33 - 4.10i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.03 - 3.37i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (0.885 + 1.53i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.06 - 3.14i)T + (31.4 + 34.9i)T^{2} \)
53 \( 1 + (-5.94 + 4.31i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (9.23 + 10.2i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (4.22 - 4.68i)T + (-6.37 - 60.6i)T^{2} \)
67 \( 1 + (11.0 - 4.93i)T + (44.8 - 49.7i)T^{2} \)
71 \( 1 + (8.02 - 5.83i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.246 - 0.759i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.222 + 0.0991i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (0.483 - 4.60i)T + (-81.1 - 17.2i)T^{2} \)
89 \( 1 + (-2.11 - 6.50i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (4.33 + 1.92i)T + (64.9 + 72.0i)T^{2} \)
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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

−11.82418978686279685662209758579, −11.09756920526833576973564945621, −10.60311647867855886246500576114, −9.271359664843675290861638043969, −7.45783278826725753016108307771, −6.62354835608186483590053258433, −5.60778130114999666000067779450, −3.87349586321894291982895664940, −3.01809355391611524042620615118, −1.45564699347577178876987318257, 3.29571664576296806699413955299, 4.46492591349898220530053989014, 5.53396359985094463738613871290, 5.87023415737069899139650130625, 7.38275718109876074997030803014, 8.890119897395006707014912134905, 9.198659779937404737524282827320, 10.75020590057470641816442574951, 12.12409181265247688966266818175, 12.73898230210177860333902714134

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