Properties

Label 2-588-196.187-c1-0-36
Degree $2$
Conductor $588$
Sign $0.981 - 0.190i$
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  + (1.32 + 0.486i)2-s + (−0.988 − 0.149i)3-s + (1.52 + 1.29i)4-s + (1.78 − 0.700i)5-s + (−1.24 − 0.678i)6-s + (−0.922 − 2.47i)7-s + (1.39 + 2.45i)8-s + (0.955 + 0.294i)9-s + (2.71 − 0.0618i)10-s + (0.216 + 0.701i)11-s + (−1.31 − 1.50i)12-s + (3.31 + 0.757i)13-s + (−0.0185 − 3.74i)14-s + (−1.86 + 0.426i)15-s + (0.661 + 3.94i)16-s + (1.16 − 1.70i)17-s + ⋯
L(s)  = 1  + (0.938 + 0.344i)2-s + (−0.570 − 0.0860i)3-s + (0.763 + 0.646i)4-s + (0.798 − 0.313i)5-s + (−0.506 − 0.277i)6-s + (−0.348 − 0.937i)7-s + (0.494 + 0.869i)8-s + (0.318 + 0.0982i)9-s + (0.857 − 0.0195i)10-s + (0.0652 + 0.211i)11-s + (−0.380 − 0.434i)12-s + (0.920 + 0.210i)13-s + (−0.00495 − 0.999i)14-s + (−0.482 + 0.110i)15-s + (0.165 + 0.986i)16-s + (0.282 − 0.414i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.981 - 0.190i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.981 - 0.190i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.49231 + 0.239466i\)
\(L(\frac12)\) \(\approx\) \(2.49231 + 0.239466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 - 0.486i)T \)
3 \( 1 + (0.988 + 0.149i)T \)
7 \( 1 + (0.922 + 2.47i)T \)
good5 \( 1 + (-1.78 + 0.700i)T + (3.66 - 3.40i)T^{2} \)
11 \( 1 + (-0.216 - 0.701i)T + (-9.08 + 6.19i)T^{2} \)
13 \( 1 + (-3.31 - 0.757i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (-1.16 + 1.70i)T + (-6.21 - 15.8i)T^{2} \)
19 \( 1 + (-2.48 + 4.29i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.444 - 0.651i)T + (-8.40 + 21.4i)T^{2} \)
29 \( 1 + (2.50 - 1.20i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-2.18 - 3.78i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.356 + 4.75i)T + (-36.5 - 5.51i)T^{2} \)
41 \( 1 + (-4.70 - 3.75i)T + (9.12 + 39.9i)T^{2} \)
43 \( 1 + (4.08 - 3.26i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (1.29 + 1.20i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (0.300 + 4.00i)T + (-52.4 + 7.89i)T^{2} \)
59 \( 1 + (0.108 - 0.275i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (12.5 + 0.943i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (-1.82 + 1.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.53 - 9.42i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (5.22 + 5.63i)T + (-5.45 + 72.7i)T^{2} \)
79 \( 1 + (7.78 + 4.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.07 - 9.07i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-2.61 + 8.48i)T + (-73.5 - 50.1i)T^{2} \)
97 \( 1 + 8.95iT - 97T^{2} \)
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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

−10.96605619747698195948665676222, −9.984276571446256418721396706678, −9.022486398350664994308503339519, −7.66610755038011629309984899565, −6.89258087619779565142330236748, −6.10238286991447293611992924163, −5.22579511919396971443189500928, −4.30716748020008662087521358394, −3.13302054215093573401343575707, −1.44808172110484889713273663129, 1.59739540004143708891298882206, 2.88038390237626766031324929739, 3.97518198193428449903408232991, 5.37935534061992552164410826831, 5.96662841368580118939609882875, 6.45460311619653815142913297909, 7.88620178186852588994291285684, 9.241703757499676468154100707321, 10.07378203666247237315304241689, 10.72689540267293438143604137058

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