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Properties

Label 2-45-9.2-c4-0-11
Degree $2$
Conductor $45$
Sign $-0.469 + 0.882i$
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  + (−3.37 − 1.94i)2-s + (8.40 − 3.22i)3-s + (−0.413 − 0.716i)4-s + (9.68 − 5.59i)5-s + (−34.6 − 5.48i)6-s + (15.5 − 26.8i)7-s + 65.5i·8-s + (60.1 − 54.1i)9-s − 43.5·10-s + (−127. − 73.7i)11-s + (−5.78 − 4.68i)12-s + (−159. − 276. i)13-s + (−104. + 60.4i)14-s + (63.3 − 78.1i)15-s + (121. − 209. i)16-s + 295. i·17-s + ⋯
L(s)  = 1  + (−0.843 − 0.486i)2-s + (0.933 − 0.358i)3-s + (−0.0258 − 0.0447i)4-s + (0.387 − 0.223i)5-s + (−0.961 − 0.152i)6-s + (0.316 − 0.548i)7-s + 1.02i·8-s + (0.743 − 0.669i)9-s − 0.435·10-s + (−1.05 − 0.609i)11-s + (−0.0401 − 0.0325i)12-s + (−0.943 − 1.63i)13-s + (−0.534 + 0.308i)14-s + (0.281 − 0.347i)15-s + (0.472 − 0.818i)16-s + 1.02i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.469 + 0.882i$
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: \((2,\ 45,\ (\ :2),\ -0.469 + 0.882i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.630005 - 1.04861i\)
\(L(\frac12)\) \(\approx\) \(0.630005 - 1.04861i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-8.40 + 3.22i)T \)
5 \( 1 + (-9.68 + 5.59i)T \)
good2 \( 1 + (3.37 + 1.94i)T + (8 + 13.8i)T^{2} \)
7 \( 1 + (-15.5 + 26.8i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (127. + 73.7i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (159. + 276. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 295. iT - 8.35e4T^{2} \)
19 \( 1 - 492.T + 1.30e5T^{2} \)
23 \( 1 + (224. - 129. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-1.07e3 - 623. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-419. - 727. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 32.2T + 1.87e6T^{2} \)
41 \( 1 + (-1.33e3 + 768. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-245. + 424. i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (1.52e3 + 882. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 2.64e3iT - 7.89e6T^{2} \)
59 \( 1 + (-2.76e3 + 1.59e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-796. + 1.38e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-2.87e3 - 4.97e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 1.43e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.95e3T + 2.83e7T^{2} \)
79 \( 1 + (-712. + 1.23e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-3.07e3 - 1.77e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 533. iT - 6.27e7T^{2} \)
97 \( 1 + (5.85e3 - 1.01e4i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−14.50045825045550299646350762810, −13.61603837649112577636814551361, −12.44062715234905261721136747062, −10.57030852352290951355952463559, −9.919075987613912069142282694945, −8.461438905556786313248350031121, −7.69338526026796953216171987364, −5.34109561216233450141940106870, −2.80862451439150397576055611001, −0.989572658130017268181016615418, 2.49204270686948438772046581896, 4.68983542945666183381175613313, 7.07535190813924974163490554595, 8.058102520709626509335320489753, 9.414743371981841090517127140992, 9.894178971263851465474267335781, 11.92919066104762939826987212770, 13.43739580864849158493128682444, 14.44019179036187018166672879272, 15.67387519301783873349246545112

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