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Properties

Label 2-15e2-9.4-c3-0-50
Degree $2$
Conductor $225$
Sign $0.401 - 0.915i$
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  + (−1.23 − 2.13i)2-s + (−4.97 − 1.50i)3-s + (0.964 − 1.67i)4-s + (2.91 + 12.4i)6-s + (−9.62 − 16.6i)7-s − 24.4·8-s + (22.4 + 14.9i)9-s + (−19.9 − 34.5i)11-s + (−7.31 + 6.85i)12-s + (0.504 − 0.873i)13-s + (−23.7 + 41.0i)14-s + (22.4 + 38.8i)16-s − 52.6·17-s + (4.27 − 66.3i)18-s − 49.5·19-s + ⋯
L(s)  = 1  + (−0.435 − 0.754i)2-s + (−0.957 − 0.289i)3-s + (0.120 − 0.208i)4-s + (0.198 + 0.848i)6-s + (−0.519 − 0.899i)7-s − 1.08·8-s + (0.832 + 0.554i)9-s + (−0.546 − 0.946i)11-s + (−0.175 + 0.164i)12-s + (0.0107 − 0.0186i)13-s + (−0.452 + 0.783i)14-s + (0.350 + 0.606i)16-s − 0.750·17-s + (0.0559 − 0.869i)18-s − 0.598·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 - 0.915i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.401 - 0.915i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.401 - 0.915i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0737921 + 0.0482094i\)
\(L(\frac12)\) \(\approx\) \(0.0737921 + 0.0482094i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.97 + 1.50i)T \)
5 \( 1 \)
good2 \( 1 + (1.23 + 2.13i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (9.62 + 16.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (19.9 + 34.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-0.504 + 0.873i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 52.6T + 4.91e3T^{2} \)
19 \( 1 + 49.5T + 6.85e3T^{2} \)
23 \( 1 + (-13.7 + 23.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-127. - 220. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-84.3 + 146. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 419.T + 5.06e4T^{2} \)
41 \( 1 + (199. - 345. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-179. - 310. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-70.6 - 122. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 290.T + 1.48e5T^{2} \)
59 \( 1 + (14.3 - 24.8i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (366. + 634. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (88.2 - 152. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 802.T + 3.57e5T^{2} \)
73 \( 1 + 512.T + 3.89e5T^{2} \)
79 \( 1 + (306. + 530. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-40.4 - 69.9i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 24.0T + 7.04e5T^{2} \)
97 \( 1 + (-683. - 1.18e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.89735679985892939479682905308, −10.42472697191784918652398656951, −9.406180767791834722716332339885, −8.034067881708877172963182937977, −6.65609016270179929368106384591, −6.01994195480656443802269101089, −4.54393280445425376959587780920, −2.86185533741461312192352995492, −1.14477879383192714323109447840, −0.05280522994184378375983033762, 2.55272564442969762897050330203, 4.36434515910954338744281045046, 5.69999014533522002862810133817, 6.48589764336261862164020428691, 7.40596595370018526123962063662, 8.668417199881110391916828787931, 9.564446256182239918764725658939, 10.55385920773368884212814236540, 11.81829969066391004114288935197, 12.32462949139224757134367048578

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