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Properties

Label 2-45-45.38-c1-0-0
Degree $2$
Conductor $45$
Sign $-0.455 - 0.890i$
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  + (−1.60 + 0.430i)2-s + (−1.08 + 1.35i)3-s + (0.661 − 0.382i)4-s + (−1.24 + 1.85i)5-s + (1.15 − 2.63i)6-s + (0.465 + 1.73i)7-s + (1.45 − 1.45i)8-s + (−0.661 − 2.92i)9-s + (1.20 − 3.51i)10-s + (3.12 + 1.80i)11-s + (−0.198 + 1.30i)12-s + (0.342 − 1.27i)13-s + (−1.49 − 2.59i)14-s + (−1.15 − 3.69i)15-s + (−2.47 + 4.28i)16-s + (−0.277 − 0.277i)17-s + ⋯
L(s)  = 1  + (−1.13 + 0.304i)2-s + (−0.624 + 0.781i)3-s + (0.330 − 0.191i)4-s + (−0.558 + 0.829i)5-s + (0.471 − 1.07i)6-s + (0.175 + 0.656i)7-s + (0.513 − 0.513i)8-s + (−0.220 − 0.975i)9-s + (0.381 − 1.11i)10-s + (0.942 + 0.544i)11-s + (−0.0573 + 0.377i)12-s + (0.0950 − 0.354i)13-s + (−0.399 − 0.692i)14-s + (−0.299 − 0.954i)15-s + (−0.618 + 1.07i)16-s + (−0.0671 − 0.0671i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.455 - 0.890i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :1/2),\ -0.455 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189266 + 0.309287i\)
\(L(\frac12)\) \(\approx\) \(0.189266 + 0.309287i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.08 - 1.35i)T \)
5 \( 1 + (1.24 - 1.85i)T \)
good2 \( 1 + (1.60 - 0.430i)T + (1.73 - i)T^{2} \)
7 \( 1 + (-0.465 - 1.73i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3.12 - 1.80i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.342 + 1.27i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.277 + 0.277i)T + 17iT^{2} \)
19 \( 1 - 6.25iT - 19T^{2} \)
23 \( 1 + (-2.16 - 0.579i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.56 + 2.71i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.42 + 4.20i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.55 + 5.55i)T - 37iT^{2} \)
41 \( 1 + (-1.29 + 0.744i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.10 + 1.10i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (3.82 - 1.02i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (7.48 - 7.48i)T - 53iT^{2} \)
59 \( 1 + (0.279 + 0.483i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.96 - 5.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.8 - 2.90i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 8.01iT - 71T^{2} \)
73 \( 1 + (1.29 + 1.29i)T + 73iT^{2} \)
79 \( 1 + (6.96 + 4.02i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.150 + 0.560i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 + (1.37 + 5.14i)T + (-84.0 + 48.5i)T^{2} \)
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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

−16.38834156846371629903760287786, −15.39920808406335687945735858078, −14.53344572241206494435041750329, −12.33813094589998815019163773622, −11.21034834923277374846623315088, −10.10714816390662001385579190722, −9.097425269889251913865302158348, −7.66167354741088512257800433577, −6.19018051018682540099364219384, −4.03074218264799297799391964867, 1.05222706323018633244862708778, 4.79838109536486840984948802532, 6.91173020135586799611166266099, 8.188763433535154635786609336300, 9.217580658652879587567557510424, 10.94373834891778675237181115080, 11.59618456214797386683551588260, 13.01599154683985898718344117318, 14.12787949575164915846621363109, 16.16550312091032499564447683183

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