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Properties

Label 2-578-17.15-c1-0-10
Degree $2$
Conductor $578$
Sign $-0.700 - 0.713i$
Analytic cond. $4.61535$
Root an. cond. $2.14833$
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  + (0.707 + 0.707i)2-s + (1.23 + 2.98i)3-s + 1.00i·4-s + (1.09 − 0.453i)5-s + (−1.23 + 2.98i)6-s + (1.03 + 0.428i)7-s + (−0.707 + 0.707i)8-s + (−5.24 + 5.24i)9-s + (1.09 + 0.453i)10-s + (1.30 − 3.15i)11-s + (−2.98 + 1.23i)12-s − 0.347i·13-s + (0.428 + 1.03i)14-s + (2.70 + 2.70i)15-s − 1.00·16-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.712 + 1.72i)3-s + 0.500i·4-s + (0.489 − 0.202i)5-s + (−0.504 + 1.21i)6-s + (0.391 + 0.162i)7-s + (−0.250 + 0.250i)8-s + (−1.74 + 1.74i)9-s + (0.346 + 0.143i)10-s + (0.393 − 0.950i)11-s + (−0.860 + 0.356i)12-s − 0.0963i·13-s + (0.114 + 0.276i)14-s + (0.697 + 0.697i)15-s − 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(578\)    =    \(2 \cdot 17^{2}\)
Sign: $-0.700 - 0.713i$
Analytic conductor: \(4.61535\)
Root analytic conductor: \(2.14833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{578} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 578,\ (\ :1/2),\ -0.700 - 0.713i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.987020 + 2.35273i\)
\(L(\frac12)\) \(\approx\) \(0.987020 + 2.35273i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 \)
good3 \( 1 + (-1.23 - 2.98i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.09 + 0.453i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.03 - 0.428i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.30 + 3.15i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 0.347iT - 13T^{2} \)
19 \( 1 + (0.245 + 0.245i)T + 19iT^{2} \)
23 \( 1 + (0.157 - 0.380i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-8.11 + 3.36i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (3.33 + 8.05i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (0.182 + 0.439i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-2.43 - 1.00i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-6.59 + 6.59i)T - 43iT^{2} \)
47 \( 1 - 7.86iT - 47T^{2} \)
53 \( 1 + (5.95 + 5.95i)T + 53iT^{2} \)
59 \( 1 + (4.53 - 4.53i)T - 59iT^{2} \)
61 \( 1 + (-5.26 - 2.18i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 7.31T + 67T^{2} \)
71 \( 1 + (-2.90 - 7.01i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-8.35 + 3.46i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (5.08 - 12.2i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (5.47 + 5.47i)T + 83iT^{2} \)
89 \( 1 - 7.18iT - 89T^{2} \)
97 \( 1 + (0.204 - 0.0848i)T + (68.5 - 68.5i)T^{2} \)
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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

−11.00521694835169915870890918505, −9.978086810779799998671313841146, −9.249359027894918891700518771489, −8.546532617134889892217128074178, −7.78171546836146363321025227488, −6.09935915070914886895790578875, −5.38851482409950620413540176825, −4.43402605327745635635994676333, −3.62725918951136475440017973104, −2.51657606880365086343024453697, 1.33123065330049806594702394158, 2.16213179295176392496494777320, 3.20283636303978275213859674883, 4.68119879198931877131301914142, 6.07588202352922342024279045084, 6.77107352762754609172848190292, 7.57559639426633189718633182346, 8.559172559502662053422153573902, 9.440495935070916529411649081358, 10.50571866615681876815696574946

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