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Properties

Label 2-588-196.55-c1-0-49
Degree $2$
Conductor $588$
Sign $-0.432 - 0.901i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
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  + (−0.712 − 1.22i)2-s + (−0.900 − 0.433i)3-s + (−0.985 + 1.74i)4-s + (1.07 − 2.23i)5-s + (0.111 + 1.40i)6-s + (−2.56 − 0.629i)7-s + (2.82 − 0.0359i)8-s + (0.623 + 0.781i)9-s + (−3.50 + 0.277i)10-s + (1.52 + 1.21i)11-s + (1.64 − 1.14i)12-s + (−3.11 − 2.48i)13-s + (1.06 + 3.58i)14-s + (−1.94 + 1.54i)15-s + (−2.05 − 3.42i)16-s + (−6.88 − 1.57i)17-s + ⋯
L(s)  = 1  + (−0.503 − 0.863i)2-s + (−0.520 − 0.250i)3-s + (−0.492 + 0.870i)4-s + (0.482 − 1.00i)5-s + (0.0455 + 0.575i)6-s + (−0.971 − 0.237i)7-s + (0.999 − 0.0126i)8-s + (0.207 + 0.260i)9-s + (−1.10 + 0.0877i)10-s + (0.458 + 0.365i)11-s + (0.474 − 0.329i)12-s + (−0.863 − 0.688i)13-s + (0.283 + 0.958i)14-s + (−0.501 + 0.400i)15-s + (−0.514 − 0.857i)16-s + (−1.67 − 0.381i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.432 - 0.901i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.432 - 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.112734 + 0.179070i\)
\(L(\frac12)\) \(\approx\) \(0.112734 + 0.179070i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.712 + 1.22i)T \)
3 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (2.56 + 0.629i)T \)
good5 \( 1 + (-1.07 + 2.23i)T + (-3.11 - 3.90i)T^{2} \)
11 \( 1 + (-1.52 - 1.21i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (3.11 + 2.48i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (6.88 + 1.57i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 - 0.903T + 19T^{2} \)
23 \( 1 + (-2.37 + 0.542i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (0.897 - 3.93i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 6.12T + 31T^{2} \)
37 \( 1 + (0.235 - 1.03i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (4.84 - 10.0i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (1.03 + 2.15i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (1.73 - 2.17i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (1.09 + 4.79i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (3.03 - 1.46i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (6.43 + 1.46i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 5.78iT - 67T^{2} \)
71 \( 1 + (0.509 - 0.116i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (6.52 - 5.20i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 13.7iT - 79T^{2} \)
83 \( 1 + (-8.67 - 10.8i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-10.0 + 8.01i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + 17.0iT - 97T^{2} \)
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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

−10.02513694466590438820278282539, −9.345260447993462006968507604962, −8.736389759698114002063063280771, −7.42609050889094518827468485113, −6.63359954424991617444116993868, −5.22055297883421290595482678317, −4.43480066476735996777469056921, −2.99011829577983406488652150738, −1.60944208141182365703497538938, −0.14575352034287110126150698695, 2.22692109004046721816545737478, 3.85916924509389737714268917645, 5.11961187990244047083588901164, 6.28421910940507490492996746935, 6.58394320955640350624010438007, 7.41517021672045136573172293717, 9.015143129111105942459973395177, 9.336072904264273281919496758651, 10.37082723672871027599893425029, 10.89655245020536387521357473960

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