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Properties

Label 2-588-196.55-c1-0-21
Degree $2$
Conductor $588$
Sign $0.689 - 0.724i$
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  + (0.311 + 1.37i)2-s + (−0.900 − 0.433i)3-s + (−1.80 + 0.858i)4-s + (0.477 − 0.992i)5-s + (0.318 − 1.37i)6-s + (−2.27 − 1.35i)7-s + (−1.74 − 2.22i)8-s + (0.623 + 0.781i)9-s + (1.51 + 0.350i)10-s + (1.30 + 1.03i)11-s + (1.99 + 0.0103i)12-s + (2.97 + 2.36i)13-s + (1.16 − 3.55i)14-s + (−0.860 + 0.686i)15-s + (2.52 − 3.10i)16-s + (4.80 + 1.09i)17-s + ⋯
L(s)  = 1  + (0.219 + 0.975i)2-s + (−0.520 − 0.250i)3-s + (−0.903 + 0.429i)4-s + (0.213 − 0.443i)5-s + (0.129 − 0.562i)6-s + (−0.858 − 0.512i)7-s + (−0.617 − 0.786i)8-s + (0.207 + 0.260i)9-s + (0.479 + 0.110i)10-s + (0.392 + 0.313i)11-s + (0.577 + 0.00299i)12-s + (0.823 + 0.657i)13-s + (0.311 − 0.950i)14-s + (−0.222 + 0.177i)15-s + (0.631 − 0.775i)16-s + (1.16 + 0.266i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.689 - 0.724i$
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: \((2,\ 588,\ (\ :1/2),\ 0.689 - 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16106 + 0.497751i\)
\(L(\frac12)\) \(\approx\) \(1.16106 + 0.497751i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.311 - 1.37i)T \)
3 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (2.27 + 1.35i)T \)
good5 \( 1 + (-0.477 + 0.992i)T + (-3.11 - 3.90i)T^{2} \)
11 \( 1 + (-1.30 - 1.03i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-2.97 - 2.36i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-4.80 - 1.09i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 - 4.75T + 19T^{2} \)
23 \( 1 + (-0.687 + 0.156i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-1.26 + 5.54i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 0.923T + 31T^{2} \)
37 \( 1 + (-2.30 + 10.1i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (1.93 - 4.01i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (1.07 + 2.22i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-2.15 + 2.69i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-2.03 - 8.93i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (10.9 - 5.29i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (1.46 + 0.333i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 3.80iT - 67T^{2} \)
71 \( 1 + (2.52 - 0.575i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-9.44 + 7.53i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 9.75iT - 79T^{2} \)
83 \( 1 + (-10.3 - 12.9i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (0.468 - 0.373i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + 5.63iT - 97T^{2} \)
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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

−10.72027195142378934315053014972, −9.618009607387114061406881752003, −9.144885625612019405505551288200, −7.86691618570023230661444460734, −7.12464586301672985527981617073, −6.23166266365974129035389842358, −5.55843677424102583030267788191, −4.38772651524244968350735280239, −3.41600193114567969831576172909, −1.02839585763079677841868044613, 1.06345531017451076863854375131, 2.97859753167880175374129288879, 3.49634058920672644329154110504, 5.03393413321206741082140935436, 5.81975666326558779669173434871, 6.65264833470041643935915019503, 8.191954387429152586676196350835, 9.204440009054664650188699700315, 9.954322227182100315659029816478, 10.54379162305627601075606417337

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