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Properties

Label 2-1044-29.4-c1-0-1
Degree $2$
Conductor $1044$
Sign $0.0870 - 0.996i$
Analytic cond. $8.33638$
Root an. cond. $2.88727$
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  + (−0.0963 + 0.422i)5-s + (−1.75 + 0.843i)7-s + (1.32 − 1.05i)11-s + (−0.887 − 1.11i)13-s + 5.95i·17-s + (0.940 − 1.95i)19-s + (1.63 + 7.16i)23-s + (4.33 + 2.08i)25-s + (−3.16 + 4.35i)29-s + (2.38 + 0.543i)31-s + (−0.187 − 0.820i)35-s + (−2.79 − 2.22i)37-s + 5.11i·41-s + (−7.35 + 1.67i)43-s + (−1.35 + 1.07i)47-s + ⋯
L(s)  = 1  + (−0.0430 + 0.188i)5-s + (−0.661 + 0.318i)7-s + (0.398 − 0.317i)11-s + (−0.246 − 0.308i)13-s + 1.44i·17-s + (0.215 − 0.447i)19-s + (0.340 + 1.49i)23-s + (0.867 + 0.417i)25-s + (−0.587 + 0.809i)29-s + (0.427 + 0.0976i)31-s + (−0.0316 − 0.138i)35-s + (−0.459 − 0.366i)37-s + 0.798i·41-s + (−1.12 + 0.255i)43-s + (−0.197 + 0.157i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0870 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0870 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $0.0870 - 0.996i$
Analytic conductor: \(8.33638\)
Root analytic conductor: \(2.88727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1044} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :1/2),\ 0.0870 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.206697738\)
\(L(\frac12)\) \(\approx\) \(1.206697738\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
29 \( 1 + (3.16 - 4.35i)T \)
good5 \( 1 + (0.0963 - 0.422i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (1.75 - 0.843i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (-1.32 + 1.05i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (0.887 + 1.11i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 - 5.95iT - 17T^{2} \)
19 \( 1 + (-0.940 + 1.95i)T + (-11.8 - 14.8i)T^{2} \)
23 \( 1 + (-1.63 - 7.16i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (-2.38 - 0.543i)T + (27.9 + 13.4i)T^{2} \)
37 \( 1 + (2.79 + 2.22i)T + (8.23 + 36.0i)T^{2} \)
41 \( 1 - 5.11iT - 41T^{2} \)
43 \( 1 + (7.35 - 1.67i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (1.35 - 1.07i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (2.56 - 11.2i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 + (-0.325 - 0.675i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (-4.67 + 5.86i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (0.751 + 0.942i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (3.72 - 0.849i)T + (65.7 - 31.6i)T^{2} \)
79 \( 1 + (-6.97 - 5.56i)T + (17.5 + 77.0i)T^{2} \)
83 \( 1 + (9.33 + 4.49i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-10.1 - 2.30i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + (-4.81 + 9.99i)T + (-60.4 - 75.8i)T^{2} \)
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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

−10.06585984208456649071958462433, −9.278065991815853790483501717980, −8.578159276817653108255830540457, −7.57259061687591105046272272733, −6.70861139133147362998302067085, −5.91560746403382285241248431435, −5.01516784191956477926387630344, −3.68457896738739225948014975359, −2.99055089632553437190060720207, −1.45643001246388010102468428769, 0.56692909631034911968663101719, 2.27457101416794164248767847602, 3.41147312520551660332719892729, 4.48077869321831658339425041661, 5.29633441688510975505764372639, 6.65564883067092226250764899376, 6.93603232392449705720722614305, 8.133374972371665641770594029182, 8.963658896462347166608532396149, 9.796284790280051102600141952170

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