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Properties

Label 2-15e2-45.34-c1-0-0
Degree $2$
Conductor $225$
Sign $-0.565 - 0.824i$
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  + (−0.495 − 0.285i)2-s + (−1.70 − 0.285i)3-s + (−0.836 − 1.44i)4-s + (0.764 + 0.630i)6-s + (−1.23 − 0.714i)7-s + 2.10i·8-s + (2.83 + 0.977i)9-s + (−1.33 + 2.31i)11-s + (1.01 + 2.71i)12-s + (−4.04 + 2.33i)13-s + (0.408 + 0.707i)14-s + (−1.07 + 1.85i)16-s + 2.67i·17-s + (−1.12 − 1.29i)18-s − 4.67·19-s + ⋯
L(s)  = 1  + (−0.350 − 0.202i)2-s + (−0.986 − 0.165i)3-s + (−0.418 − 0.724i)4-s + (0.312 + 0.257i)6-s + (−0.467 − 0.269i)7-s + 0.742i·8-s + (0.945 + 0.325i)9-s + (−0.402 + 0.697i)11-s + (0.292 + 0.783i)12-s + (−1.12 + 0.648i)13-s + (0.109 + 0.189i)14-s + (−0.267 + 0.464i)16-s + 0.648i·17-s + (−0.265 − 0.305i)18-s − 1.07·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.565 - 0.824i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.565 - 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0412628 + 0.0783057i\)
\(L(\frac12)\) \(\approx\) \(0.0412628 + 0.0783057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 + 0.285i)T \)
5 \( 1 \)
good2 \( 1 + (0.495 + 0.285i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (1.23 + 0.714i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.33 - 2.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.04 - 2.33i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.67iT - 17T^{2} \)
19 \( 1 + 4.67T + 19T^{2} \)
23 \( 1 + (-5.12 + 2.95i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.74 - 8.21i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.48 + 6.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.81iT - 37T^{2} \)
41 \( 1 + (-0.735 - 1.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.408 - 0.235i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.02 + 3.47i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.14iT - 53T^{2} \)
59 \( 1 + (0.571 + 0.990i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.26 + 2.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.70 - 3.29i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 - 1.71iT - 73T^{2} \)
79 \( 1 + (0.143 - 0.249i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.71 + 2.14i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (6.78 + 3.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.68555663391719179951156126051, −11.40107957610659832454538645099, −10.56234938887180853812903070166, −9.916049306508257672284360568073, −8.941607289859662366273834159564, −7.38602051231870354288663445135, −6.46900502512227659582461474997, −5.26468119922338314541740162259, −4.37971992086850272606877256482, −1.91912018750320778659264059452, 0.089625988368504343775529566565, 3.10011795885534956160573264503, 4.57745458865628653567726890710, 5.67160354249808067054995191971, 6.92056610972392828954913845849, 7.81898512002321956233472310169, 9.096005978355202727248146158325, 9.889811303182592396223879625102, 10.95724754823836543799657851218, 11.97615453576407690212822351983

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