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Properties

Label 2-578-17.8-c1-0-10
Degree $2$
Conductor $578$
Sign $0.946 + 0.323i$
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 + (0.765 − 1.84i)3-s − 1.00i·4-s + (0.765 + 1.84i)6-s + (3.69 − 1.53i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (2.29 + 5.54i)11-s + (−1.84 − 0.765i)12-s + 2i·13-s + (−1.53 + 3.69i)14-s − 1.00·16-s + 1.00·18-s + (2.82 − 2.82i)19-s − 8i·21-s + (−5.54 − 2.29i)22-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.441 − 1.06i)3-s − 0.500i·4-s + (0.312 + 0.754i)6-s + (1.39 − 0.578i)7-s + (0.250 + 0.250i)8-s + (−0.235 − 0.235i)9-s + (0.692 + 1.67i)11-s + (−0.533 − 0.220i)12-s + 0.554i·13-s + (−0.409 + 0.987i)14-s − 0.250·16-s + 0.235·18-s + (0.648 − 0.648i)19-s − 1.74i·21-s + (−1.18 − 0.489i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(578\)    =    \(2 \cdot 17^{2}\)
Sign: $0.946 + 0.323i$
Analytic conductor: \(4.61535\)
Root analytic conductor: \(2.14833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{578} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 578,\ (\ :1/2),\ 0.946 + 0.323i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57478 - 0.261476i\)
\(L(\frac12)\) \(\approx\) \(1.57478 - 0.261476i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.765 + 1.84i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-3.69 + 1.53i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-2.29 - 5.54i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
19 \( 1 + (-2.82 + 2.82i)T - 19iT^{2} \)
23 \( 1 + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (1.53 - 3.69i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.53 + 3.69i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (5.54 - 2.29i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (5.65 + 5.65i)T + 43iT^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + (-4.24 + 4.24i)T - 53iT^{2} \)
59 \( 1 + 59iT^{2} \)
61 \( 1 + (3.69 - 1.53i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.84 + 0.765i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.06 - 7.39i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (-12.9 - 5.35i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.55526386509175712605492465738, −9.644280112006844643503334556599, −8.662455441986952871872382603042, −7.83392544749716840054080629891, −7.19562733220390737787056021633, −6.69655898075658681638170320445, −5.08296674128438125958459953122, −4.25396918588014835502066789728, −2.09804999257561438455093869979, −1.39141994190132169982894463692, 1.42147892705082335695064735520, 3.05430478848587715500267968355, 3.85769980144400858831983875865, 5.03080387762314304020861156460, 5.99767177604757715921228471612, 7.68828306853756778607409806728, 8.435129330643874687712747995398, 8.996308187277015557971012467114, 9.859630424823319727152088328751, 10.74550813667282175776454746650

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