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Properties

Label 2-592-148.103-c1-0-9
Degree $2$
Conductor $592$
Sign $0.999 + 0.0294i$
Analytic cond. $4.72714$
Root an. cond. $2.17419$
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  + (0.436 − 0.756i)3-s + (−0.192 − 0.717i)5-s + (1.90 + 1.09i)7-s + (1.11 + 1.93i)9-s + 1.68·11-s + (1.56 + 5.83i)13-s + (−0.626 − 0.167i)15-s + (−4.60 − 1.23i)17-s + (3.93 − 1.05i)19-s + (1.66 − 0.959i)21-s + (−5.76 − 5.76i)23-s + (3.85 − 2.22i)25-s + 4.57·27-s + (5.98 + 5.98i)29-s + (7.55 − 7.55i)31-s + ⋯
L(s)  = 1  + (0.252 − 0.436i)3-s + (−0.0859 − 0.320i)5-s + (0.719 + 0.415i)7-s + (0.372 + 0.645i)9-s + 0.509·11-s + (0.433 + 1.61i)13-s + (−0.161 − 0.0433i)15-s + (−1.11 − 0.299i)17-s + (0.901 − 0.241i)19-s + (0.362 − 0.209i)21-s + (−1.20 − 1.20i)23-s + (0.770 − 0.444i)25-s + 0.880·27-s + (1.11 + 1.11i)29-s + (1.35 − 1.35i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $0.999 + 0.0294i$
Analytic conductor: \(4.72714\)
Root analytic conductor: \(2.17419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 592,\ (\ :1/2),\ 0.999 + 0.0294i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79524 - 0.0264276i\)
\(L(\frac12)\) \(\approx\) \(1.79524 - 0.0264276i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (5.44 - 2.71i)T \)
good3 \( 1 + (-0.436 + 0.756i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.192 + 0.717i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.90 - 1.09i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.68T + 11T^{2} \)
13 \( 1 + (-1.56 - 5.83i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.60 + 1.23i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.93 + 1.05i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.76 + 5.76i)T + 23iT^{2} \)
29 \( 1 + (-5.98 - 5.98i)T + 29iT^{2} \)
31 \( 1 + (-7.55 + 7.55i)T - 31iT^{2} \)
41 \( 1 + (4.64 + 2.68i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.512 + 0.512i)T + 43iT^{2} \)
47 \( 1 - 10.8iT - 47T^{2} \)
53 \( 1 + (-0.799 - 1.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.25 + 12.1i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.62 - 0.704i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.515 - 0.892i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.02 + 1.16i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.19iT - 73T^{2} \)
79 \( 1 + (11.6 - 3.12i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (5.53 - 3.19i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.03 + 3.86i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-10.1 + 10.1i)T - 97iT^{2} \)
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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

−10.80093346631669346849909721355, −9.699874624371534922618807472784, −8.637601789749005565472886225721, −8.314226776890254171501458233150, −7.00440705943964236102928460690, −6.40443126154292271603912817654, −4.83295908581897346287861142665, −4.33947739006312877504526559321, −2.50735495944985895702036906225, −1.48714734239647228450175843699, 1.24503460193354819848016318557, 3.08398800820073045842236047823, 3.94167643897108295821534603612, 5.00631473502535176320594924407, 6.18703578063666696825114584236, 7.15981941289825635358252334494, 8.150269623934758774594705577779, 8.853288548597642985352936340602, 10.11742144478889740997002323353, 10.38923774946286876432339273424

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