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Properties

Label 2-578-17.9-c1-0-16
Degree $2$
Conductor $578$
Sign $0.484 + 0.874i$
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 + (1.84 + 0.765i)3-s − 1.00i·4-s + (1.84 − 0.765i)6-s + (−1.53 − 3.69i)7-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (5.54 − 2.29i)11-s + (0.765 − 1.84i)12-s + 2i·13-s + (−3.69 − 1.53i)14-s − 1.00·16-s + 1.00·18-s + (−2.82 + 2.82i)19-s − 8i·21-s + (2.29 − 5.54i)22-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (1.06 + 0.441i)3-s − 0.500i·4-s + (0.754 − 0.312i)6-s + (−0.578 − 1.39i)7-s + (−0.250 − 0.250i)8-s + (0.235 + 0.235i)9-s + (1.67 − 0.692i)11-s + (0.220 − 0.533i)12-s + 0.554i·13-s + (−0.987 − 0.409i)14-s − 0.250·16-s + 0.235·18-s + (−0.648 + 0.648i)19-s − 1.74i·21-s + (0.489 − 1.18i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(578\)    =    \(2 \cdot 17^{2}\)
Sign: $0.484 + 0.874i$
Analytic conductor: \(4.61535\)
Root analytic conductor: \(2.14833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{578} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 578,\ (\ :1/2),\ 0.484 + 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21868 - 1.30714i\)
\(L(\frac12)\) \(\approx\) \(2.21868 - 1.30714i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.84 - 0.765i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.53 + 3.69i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-5.54 + 2.29i)T + (7.77 - 7.77i)T^{2} \)
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 \( 1 + (3.69 + 1.53i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-3.69 - 1.53i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-2.29 - 5.54i)T + (-28.9 + 28.9i)T^{2} \)
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 \( 1 + (-1.53 - 3.69i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-0.765 + 1.84i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-7.39 + 3.06i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (5.35 - 12.9i)T + (-68.5 - 68.5i)T^{2} \)
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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

−10.61749415392712310193290480581, −9.483956182754472329321204568324, −9.232429880786897619518359786402, −8.043178118001356220929000996668, −6.84987955121010103652661131455, −6.09845216799663032334335709488, −4.26999237902438684428827481811, −3.88331564778140735433892285895, −2.99907826588688147534435151785, −1.29072450440226118455906353889, 2.09596934791002315746910879496, 2.99912227779041885373800417268, 4.13761324265205908385737050625, 5.46436810315769470993088032257, 6.45347290757400571778436880560, 7.17472011681921517070257559618, 8.331482392475738344194186729150, 8.974015517082527597484707962711, 9.476780609831646083924153546797, 11.00830595408115395776069531032

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