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Properties

Label 2-45-9.2-c4-0-12
Degree $2$
Conductor $45$
Sign $0.685 + 0.728i$
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  + (2.59 + 1.49i)2-s + (4.94 − 7.51i)3-s + (−3.50 − 6.07i)4-s + (9.68 − 5.59i)5-s + (24.1 − 12.0i)6-s + (−1.63 + 2.83i)7-s − 68.9i·8-s + (−31.9 − 74.4i)9-s + 33.5·10-s + (81.9 + 47.2i)11-s + (−63.0 − 3.70i)12-s + (110. + 190. i)13-s + (−8.50 + 4.90i)14-s + (5.90 − 100. i)15-s + (47.3 − 81.9i)16-s + 169. i·17-s + ⋯
L(s)  = 1  + (0.649 + 0.374i)2-s + (0.549 − 0.835i)3-s + (−0.219 − 0.379i)4-s + (0.387 − 0.223i)5-s + (0.669 − 0.335i)6-s + (−0.0334 + 0.0578i)7-s − 1.07i·8-s + (−0.395 − 0.918i)9-s + 0.335·10-s + (0.676 + 0.390i)11-s + (−0.437 − 0.0257i)12-s + (0.651 + 1.12i)13-s + (−0.0433 + 0.0250i)14-s + (0.0262 − 0.446i)15-s + (0.184 − 0.320i)16-s + 0.586i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.685 + 0.728i$
Analytic conductor: \(4.65164\)
Root analytic conductor: \(2.15676\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :2),\ 0.685 + 0.728i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.16146 - 0.933786i\)
\(L(\frac12)\) \(\approx\) \(2.16146 - 0.933786i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.94 + 7.51i)T \)
5 \( 1 + (-9.68 + 5.59i)T \)
good2 \( 1 + (-2.59 - 1.49i)T + (8 + 13.8i)T^{2} \)
7 \( 1 + (1.63 - 2.83i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-81.9 - 47.2i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-110. - 190. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 169. iT - 8.35e4T^{2} \)
19 \( 1 - 87.9T + 1.30e5T^{2} \)
23 \( 1 + (421. - 243. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (403. + 233. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-470. - 814. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 1.34e3T + 1.87e6T^{2} \)
41 \( 1 + (2.49e3 - 1.44e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.70e3 + 2.95e3i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-1.19e3 - 691. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 4.66e3iT - 7.89e6T^{2} \)
59 \( 1 + (-2.30e3 + 1.33e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (534. - 926. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-859. - 1.48e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 96.5iT - 2.54e7T^{2} \)
73 \( 1 + 6.75e3T + 2.83e7T^{2} \)
79 \( 1 + (5.66e3 - 9.80e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (7.24e3 + 4.18e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 7.61e3iT - 6.27e7T^{2} \)
97 \( 1 + (-8.95e3 + 1.55e4i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−14.54060685690872980812055998255, −13.88119403369059883507413285519, −12.97889915327148527117882274604, −11.79480446770283711087276611403, −9.795782195282271728571608497289, −8.725518781628221967470213143665, −6.93282117794456327097628692317, −5.92426344389694797353735062005, −4.01720538491146729538278878613, −1.52574635350894693000882107260, 2.84032082504698660190503909959, 4.08130162174979787863741443523, 5.64813905294000242555609244652, 7.974314738503283834623791961911, 9.147331867399924695533505271265, 10.51583988488652075991052975397, 11.68885625613623083729085856219, 13.20901100864109475089651433991, 13.96147340038261964291930660141, 14.94878145347579429055887095094

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