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Properties

Label 2-574-41.9-c1-0-5
Degree $2$
Conductor $574$
Sign $-0.712 - 0.701i$
Analytic cond. $4.58341$
Root an. cond. $2.14089$
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 + (1.36 + 1.36i)3-s − 4-s + 1.10i·5-s + (−1.36 + 1.36i)6-s + (0.707 + 0.707i)7-s − i·8-s + 0.726i·9-s − 1.10·10-s + (3.14 + 3.14i)11-s + (−1.36 − 1.36i)12-s + (0.465 + 0.465i)13-s + (−0.707 + 0.707i)14-s + (−1.50 + 1.50i)15-s + 16-s + (−0.0491 + 0.0491i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.788 + 0.788i)3-s − 0.5·4-s + 0.492i·5-s + (−0.557 + 0.557i)6-s + (0.267 + 0.267i)7-s − 0.353i·8-s + 0.242i·9-s − 0.348·10-s + (0.948 + 0.948i)11-s + (−0.394 − 0.394i)12-s + (0.129 + 0.129i)13-s + (−0.188 + 0.188i)14-s + (−0.388 + 0.388i)15-s + 0.250·16-s + (−0.0119 + 0.0119i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(574\)    =    \(2 \cdot 7 \cdot 41\)
Sign: $-0.712 - 0.701i$
Analytic conductor: \(4.58341\)
Root analytic conductor: \(2.14089\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{574} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 574,\ (\ :1/2),\ -0.712 - 0.701i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692636 + 1.68943i\)
\(L(\frac12)\) \(\approx\) \(0.692636 + 1.68943i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
7 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-2.07 + 6.05i)T \)
good3 \( 1 + (-1.36 - 1.36i)T + 3iT^{2} \)
5 \( 1 - 1.10iT - 5T^{2} \)
11 \( 1 + (-3.14 - 3.14i)T + 11iT^{2} \)
13 \( 1 + (-0.465 - 0.465i)T + 13iT^{2} \)
17 \( 1 + (0.0491 - 0.0491i)T - 17iT^{2} \)
19 \( 1 + (3.25 - 3.25i)T - 19iT^{2} \)
23 \( 1 + 4.56T + 23T^{2} \)
29 \( 1 + (1.38 + 1.38i)T + 29iT^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 - 2.56T + 37T^{2} \)
43 \( 1 + 0.265iT - 43T^{2} \)
47 \( 1 + (-7.39 + 7.39i)T - 47iT^{2} \)
53 \( 1 + (-7.01 - 7.01i)T + 53iT^{2} \)
59 \( 1 - 5.52T + 59T^{2} \)
61 \( 1 + 9.96iT - 61T^{2} \)
67 \( 1 + (0.864 - 0.864i)T - 67iT^{2} \)
71 \( 1 + (5.54 + 5.54i)T + 71iT^{2} \)
73 \( 1 - 11.2iT - 73T^{2} \)
79 \( 1 + (-9.54 - 9.54i)T + 79iT^{2} \)
83 \( 1 - 2.94T + 83T^{2} \)
89 \( 1 + (7.87 + 7.87i)T + 89iT^{2} \)
97 \( 1 + (-5.01 + 5.01i)T - 97iT^{2} \)
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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

−10.76937782325955342267217100202, −9.950341997640452114620338839062, −9.167821717735601469847240845787, −8.581278921974096711801331001850, −7.49363297763202423234649717133, −6.64048621818897982860334842236, −5.59521107065744805305167758722, −4.24357325876981255346340249049, −3.72459296876846301490350886593, −2.12810927681282437286728671122, 1.04147747448310748272535530372, 2.18558420638471304622484682122, 3.42545956642293358588773340870, 4.46786224617919515234526536134, 5.76911231170285957775343551389, 6.97445713166057797833286959053, 7.966276658699701290019249949023, 8.764997481500287289329934186870, 9.216000569534242171078060395653, 10.59954104157279738889300949027

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