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Properties

Label 2-45-45.34-c3-0-2
Degree $2$
Conductor $45$
Sign $0.971 + 0.235i$
Analytic cond. $2.65508$
Root an. cond. $1.62944$
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  + (−4.37 − 2.52i)2-s + (5.16 − 0.559i)3-s + (8.77 + 15.1i)4-s + (−3.07 + 10.7i)5-s + (−24.0 − 10.6i)6-s + (18.1 + 10.5i)7-s − 48.2i·8-s + (26.3 − 5.78i)9-s + (40.6 − 39.2i)10-s + (14.2 − 24.7i)11-s + (53.8 + 73.5i)12-s + (−8.72 + 5.03i)13-s + (−53.1 − 91.9i)14-s + (−9.86 + 57.2i)15-s + (−51.7 + 89.6i)16-s + 82.7i·17-s + ⋯
L(s)  = 1  + (−1.54 − 0.893i)2-s + (0.994 − 0.107i)3-s + (1.09 + 1.89i)4-s + (−0.274 + 0.961i)5-s + (−1.63 − 0.721i)6-s + (0.982 + 0.567i)7-s − 2.13i·8-s + (0.976 − 0.214i)9-s + (1.28 − 1.24i)10-s + (0.391 − 0.678i)11-s + (1.29 + 1.77i)12-s + (−0.186 + 0.107i)13-s + (−1.01 − 1.75i)14-s + (−0.169 + 0.985i)15-s + (−0.808 + 1.40i)16-s + 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.971 + 0.235i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ 0.971 + 0.235i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.931515 - 0.111037i\)
\(L(\frac12)\) \(\approx\) \(0.931515 - 0.111037i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.16 + 0.559i)T \)
5 \( 1 + (3.07 - 10.7i)T \)
good2 \( 1 + (4.37 + 2.52i)T + (4 + 6.92i)T^{2} \)
7 \( 1 + (-18.1 - 10.5i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-14.2 + 24.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (8.72 - 5.03i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 82.7iT - 4.91e3T^{2} \)
19 \( 1 + 1.91T + 6.85e3T^{2} \)
23 \( 1 + (-147. + 85.2i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (128. - 222. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (24.0 + 41.6i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 161. iT - 5.06e4T^{2} \)
41 \( 1 + (139. + 241. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (233. + 134. i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (8.30 + 4.79i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 35.7iT - 1.48e5T^{2} \)
59 \( 1 + (281. + 487. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-39.6 + 68.6i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (404. - 233. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 316.T + 3.57e5T^{2} \)
73 \( 1 + 633. iT - 3.89e5T^{2} \)
79 \( 1 + (395. - 685. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (197. + 114. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 53.9T + 7.04e5T^{2} \)
97 \( 1 + (83.7 + 48.3i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−15.23009473389638818428303140810, −14.37560777165430249504754014483, −12.57480136985567176983799462363, −11.28400896395939299761280854406, −10.48766435245690953563098008852, −9.002001542131825022276991382142, −8.278055290616344394234330775528, −7.08636762711157414377615321310, −3.37222849247477998789449646729, −1.89941574549633252367207202499, 1.36650640148180238890466420988, 4.79947611430918987878509760131, 7.23978934750438407042717219405, 7.958432984064842884209053735917, 9.062615797259890767323729120588, 9.858191985058069923215373536071, 11.47194102844799984661658563814, 13.41628215194485252714799471205, 14.77666259140167627006976629698, 15.49349324606336025228444004018

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