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Properties

Label 2-578-17.13-c1-0-5
Degree $2$
Conductor $578$
Sign $-0.854 - 0.519i$
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  + i·2-s + (1.54 + 1.54i)3-s − 4-s + (2.41 + 2.41i)5-s + (−1.54 + 1.54i)6-s + (−2.36 + 2.36i)7-s − i·8-s + 1.77i·9-s + (−2.41 + 2.41i)10-s + (1.57 − 1.57i)11-s + (−1.54 − 1.54i)12-s − 1.53·13-s + (−2.36 − 2.36i)14-s + 7.45i·15-s + 16-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.891 + 0.891i)3-s − 0.5·4-s + (1.07 + 1.07i)5-s + (−0.630 + 0.630i)6-s + (−0.894 + 0.894i)7-s − 0.353i·8-s + 0.591i·9-s + (−0.762 + 0.762i)10-s + (0.474 − 0.474i)11-s + (−0.445 − 0.445i)12-s − 0.424·13-s + (−0.632 − 0.632i)14-s + 1.92i·15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 - 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(578\)    =    \(2 \cdot 17^{2}\)
Sign: $-0.854 - 0.519i$
Analytic conductor: \(4.61535\)
Root analytic conductor: \(2.14833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{578} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 578,\ (\ :1/2),\ -0.854 - 0.519i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.539798 + 1.92498i\)
\(L(\frac12)\) \(\approx\) \(0.539798 + 1.92498i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
17 \( 1 \)
good3 \( 1 + (-1.54 - 1.54i)T + 3iT^{2} \)
5 \( 1 + (-2.41 - 2.41i)T + 5iT^{2} \)
7 \( 1 + (2.36 - 2.36i)T - 7iT^{2} \)
11 \( 1 + (-1.57 + 1.57i)T - 11iT^{2} \)
13 \( 1 + 1.53T + 13T^{2} \)
19 \( 1 + 1.53iT - 19T^{2} \)
23 \( 1 + (-3.69 + 3.69i)T - 23iT^{2} \)
29 \( 1 + (4.27 + 4.27i)T + 29iT^{2} \)
31 \( 1 + (-0.501 - 0.501i)T + 31iT^{2} \)
37 \( 1 + (-8.47 - 8.47i)T + 37iT^{2} \)
41 \( 1 + (-2.85 + 2.85i)T - 41iT^{2} \)
43 \( 1 + 5.02iT - 43T^{2} \)
47 \( 1 - 0.142T + 47T^{2} \)
53 \( 1 - 13.0iT - 53T^{2} \)
59 \( 1 + 0.773iT - 59T^{2} \)
61 \( 1 + (5.51 - 5.51i)T - 61iT^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 + (1.86 + 1.86i)T + 71iT^{2} \)
73 \( 1 + (8.90 + 8.90i)T + 73iT^{2} \)
79 \( 1 + (0.0592 - 0.0592i)T - 79iT^{2} \)
83 \( 1 + 10.6iT - 83T^{2} \)
89 \( 1 + 2.58T + 89T^{2} \)
97 \( 1 + (-6.54 - 6.54i)T + 97iT^{2} \)
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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

−10.63104841964993399431297808392, −9.931581665808412181412581620775, −9.204786805326329562081408315378, −8.851724112046215034378764577398, −7.45190475120978961518855208809, −6.32897804197591368492580831348, −5.93489499618058047375973080126, −4.51533070449860995726330913436, −3.17183264374281683223300853446, −2.58573957649653540286702641020, 1.11891846513379119203771161837, 2.04802848964825673948769862248, 3.28694813340437043756474154876, 4.52302992937643620226492185819, 5.72116505967804927055449984616, 6.92900213551065845508987593696, 7.74080817250846876315969143167, 8.899658959917589401756359390496, 9.464924412424298335491374975480, 10.04164154779486141345563861521

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