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Properties

Label 2-15e2-9.7-c3-0-25
Degree $2$
Conductor $225$
Sign $0.943 + 0.330i$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.28 + 3.96i)2-s + (−3.36 − 3.96i)3-s + (−6.45 − 11.1i)4-s + (23.3 − 4.26i)6-s + (−10.0 + 17.4i)7-s + 22.4·8-s + (−4.37 + 26.6i)9-s + (−33.1 + 57.4i)11-s + (−22.5 + 63.2i)12-s + (−23.4 − 40.5i)13-s + (−45.9 − 79.6i)14-s + (0.237 − 0.411i)16-s + 47.6·17-s + (−95.5 − 78.2i)18-s − 9.95·19-s + ⋯
L(s)  = 1  + (−0.808 + 1.40i)2-s + (−0.647 − 0.762i)3-s + (−0.807 − 1.39i)4-s + (1.59 − 0.290i)6-s + (−0.543 + 0.940i)7-s + 0.993·8-s + (−0.162 + 0.986i)9-s + (−0.909 + 1.57i)11-s + (−0.543 + 1.52i)12-s + (−0.499 − 0.864i)13-s + (−0.878 − 1.52i)14-s + (0.00371 − 0.00643i)16-s + 0.679·17-s + (−1.25 − 1.02i)18-s − 0.120·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.943 + 0.330i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.943 + 0.330i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.294840 - 0.0502080i\)
\(L(\frac12)\) \(\approx\) \(0.294840 - 0.0502080i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.36 + 3.96i)T \)
5 \( 1 \)
good2 \( 1 + (2.28 - 3.96i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (10.0 - 17.4i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (33.1 - 57.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (23.4 + 40.5i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 47.6T + 4.91e3T^{2} \)
19 \( 1 + 9.95T + 6.85e3T^{2} \)
23 \( 1 + (-4.79 - 8.30i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-89.3 + 154. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (77.0 + 133. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 248.T + 5.06e4T^{2} \)
41 \( 1 + (124. + 216. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (106. - 183. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-237. + 411. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 546.T + 1.48e5T^{2} \)
59 \( 1 + (-209. - 363. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-272. + 472. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (223. + 387. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 409.T + 3.57e5T^{2} \)
73 \( 1 - 358.T + 3.89e5T^{2} \)
79 \( 1 + (-325. + 564. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (406. - 704. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 200.T + 7.04e5T^{2} \)
97 \( 1 + (-126. + 218. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.07948584025783279758560037653, −10.34334291867878602265615036077, −9.711262260817522990295617917859, −8.399678956758976257842353892285, −7.57865521139392017500206551432, −6.87703522525078749458995727091, −5.71847167918843917497419534861, −5.14328922242240772129558434686, −2.35840736866336011463408864364, −0.23413845665253988881949403473, 0.915041730627423745209140239831, 3.02319003661074136475459149154, 3.87390192234261515494197132685, 5.38143632498261329945670915146, 6.83495794378899783409763730455, 8.393751307887882064283060239565, 9.277933609410937043342933845763, 10.35658406912135711714443801951, 10.57888377435131675848121268268, 11.54251073669190235268995163896

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