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Properties

Label 2-45-45.2-c1-0-3
Degree $2$
Conductor $45$
Sign $-0.637 + 0.770i$
Analytic cond. $0.359326$
Root an. cond. $0.599438$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.601 − 2.24i)2-s + (−1.72 − 0.173i)3-s + (−2.93 + 1.69i)4-s + (1.70 − 1.44i)5-s + (0.647 + 3.97i)6-s + (0.751 − 0.201i)7-s + (2.29 + 2.29i)8-s + (2.93 + 0.597i)9-s + (−4.26 − 2.96i)10-s + (−0.220 − 0.127i)11-s + (5.36 − 2.41i)12-s + (3.70 + 0.992i)13-s + (−0.903 − 1.56i)14-s + (−3.19 + 2.18i)15-s + (0.367 − 0.636i)16-s + (−3.93 + 3.93i)17-s + ⋯
L(s)  = 1  + (−0.425 − 1.58i)2-s + (−0.994 − 0.100i)3-s + (−1.46 + 0.848i)4-s + (0.764 − 0.644i)5-s + (0.264 + 1.62i)6-s + (0.284 − 0.0761i)7-s + (0.809 + 0.809i)8-s + (0.979 + 0.199i)9-s + (−1.34 − 0.938i)10-s + (−0.0663 − 0.0383i)11-s + (1.54 − 0.697i)12-s + (1.02 + 0.275i)13-s + (−0.241 − 0.418i)14-s + (−0.825 + 0.565i)15-s + (0.0918 − 0.159i)16-s + (−0.953 + 0.953i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.637 + 0.770i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1/2),\ -0.637 + 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.235744 - 0.501129i\)
\(L(\frac12)\) \(\approx\) \(0.235744 - 0.501129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.173i)T \)
5 \( 1 + (-1.70 + 1.44i)T \)
good2 \( 1 + (0.601 + 2.24i)T + (-1.73 + i)T^{2} \)
7 \( 1 + (-0.751 + 0.201i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.220 + 0.127i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.70 - 0.992i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (3.93 - 3.93i)T - 17iT^{2} \)
19 \( 1 - 0.440iT - 19T^{2} \)
23 \( 1 + (0.917 - 3.42i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.76 + 4.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.0971 + 0.168i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.123 + 0.123i)T + 37iT^{2} \)
41 \( 1 + (3.88 - 2.24i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.357 + 1.33i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.11 - 4.17i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.938 + 0.938i)T + 53iT^{2} \)
59 \( 1 + (-4.02 - 6.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.44 - 2.49i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.47 + 12.9i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.15iT - 71T^{2} \)
73 \( 1 + (9.18 - 9.18i)T - 73iT^{2} \)
79 \( 1 + (11.9 + 6.91i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.20 - 1.39i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 0.285T + 89T^{2} \)
97 \( 1 + (-8.73 + 2.34i)T + (84.0 - 48.5i)T^{2} \)
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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

−15.79506490531667673538142258798, −13.56327587723444445145685730476, −12.85979086557917012419546876298, −11.71753712997049507383531598963, −10.82066849670382442913184624024, −9.822336784804281480910066129597, −8.544433288176183711783524410051, −6.11161414243586899283902994747, −4.33850739364639117323882646390, −1.59479151241026280775828994449, 5.05079104663487634680836234217, 6.23184092004182062177443347872, 7.06596992021042963301161596096, 8.766189348353140560387234184363, 10.13272999266800891792769009750, 11.33364972787446945031709351497, 13.22682594577075160996934153518, 14.33624213801145958768892077363, 15.51383260347101794328087745761, 16.26708285668580582087153641316

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