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Properties

Label 2-45-45.14-c4-0-12
Degree $2$
Conductor $45$
Sign $0.364 - 0.931i$
Analytic cond. $4.65164$
Root an. cond. $2.15676$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.87 + 4.97i)2-s + (8.30 − 3.46i)3-s + (−8.51 + 14.7i)4-s + (24.3 − 5.74i)5-s + (41.1 + 31.3i)6-s + (−41.4 + 23.9i)7-s − 5.92·8-s + (56.9 − 57.6i)9-s + (98.5 + 104. i)10-s + (−73.2 + 42.2i)11-s + (−19.5 + 152. i)12-s + (−193. − 111. i)13-s + (−238. − 137. i)14-s + (182. − 132. i)15-s + (119. + 206. i)16-s − 434.·17-s + ⋯
L(s)  = 1  + (0.718 + 1.24i)2-s + (0.922 − 0.385i)3-s + (−0.532 + 0.921i)4-s + (0.973 − 0.229i)5-s + (1.14 + 0.871i)6-s + (−0.845 + 0.487i)7-s − 0.0925·8-s + (0.702 − 0.711i)9-s + (0.985 + 1.04i)10-s + (−0.605 + 0.349i)11-s + (−0.135 + 1.05i)12-s + (−1.14 − 0.660i)13-s + (−1.21 − 0.701i)14-s + (0.809 − 0.587i)15-s + (0.465 + 0.806i)16-s − 1.50·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.364 - 0.931i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ 0.364 - 0.931i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.34907 + 1.60313i\)
\(L(\frac12)\) \(\approx\) \(2.34907 + 1.60313i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-8.30 + 3.46i)T \)
5 \( 1 + (-24.3 + 5.74i)T \)
good2 \( 1 + (-2.87 - 4.97i)T + (-8 + 13.8i)T^{2} \)
7 \( 1 + (41.4 - 23.9i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (73.2 - 42.2i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (193. + 111. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 434.T + 8.35e4T^{2} \)
19 \( 1 - 378.T + 1.30e5T^{2} \)
23 \( 1 + (-326. + 566. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (430. - 248. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-151. + 262. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 55.6iT - 1.87e6T^{2} \)
41 \( 1 + (-411. - 237. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (707. - 408. i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-435. - 753. i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 2.33e3T + 7.89e6T^{2} \)
59 \( 1 + (1.01e3 + 588. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-3.45e3 - 5.98e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-3.35e3 - 1.93e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 5.82e3iT - 2.54e7T^{2} \)
73 \( 1 - 6.44e3iT - 2.83e7T^{2} \)
79 \( 1 + (2.44e3 + 4.23e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (3.25e3 + 5.64e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 1.38e4iT - 6.27e7T^{2} \)
97 \( 1 + (-1.16e4 + 6.74e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−15.17097236221495509200472696224, −14.26538155306484801406837477148, −13.10441336238746113111724343147, −12.76987248561940118424403658015, −10.04805771373600066370198253980, −8.898425952008286434229766863013, −7.41603338137272613775806609986, −6.33986832834333761090560744239, −4.91993861351249030071450374584, −2.60605378969782882003718638271, 2.16556499183514916530016279322, 3.33532481142731249782305426077, 4.94949224233529713549177344203, 7.16904269329065683417058064019, 9.365396157164196868897201657928, 10.03482847102890605233084044210, 11.19173718824359503410672823080, 12.92485834256347155394045280329, 13.51497160615806488731539733808, 14.28185994816800617519230489661

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