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Properties

Label 2-4788-19.7-c1-0-29
Degree $2$
Conductor $4788$
Sign $-0.321 + 0.946i$
Analytic cond. $38.2323$
Root an. cond. $6.18323$
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  + (−1.5 − 2.59i)5-s + 7-s − 3·11-s + (2 − 3.46i)13-s + (1.5 + 2.59i)17-s + (−0.5 + 4.33i)19-s + (−2 + 3.46i)25-s + 8·31-s + (−1.5 − 2.59i)35-s + 11·37-s + (4.5 + 7.79i)41-s + (−4 − 6.92i)43-s + (6 − 10.3i)47-s + 49-s + (6 − 10.3i)53-s + ⋯
L(s)  = 1  + (−0.670 − 1.16i)5-s + 0.377·7-s − 0.904·11-s + (0.554 − 0.960i)13-s + (0.363 + 0.630i)17-s + (−0.114 + 0.993i)19-s + (−0.400 + 0.692i)25-s + 1.43·31-s + (−0.253 − 0.439i)35-s + 1.80·37-s + (0.702 + 1.21i)41-s + (−0.609 − 1.05i)43-s + (0.875 − 1.51i)47-s + 0.142·49-s + (0.824 − 1.42i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4788\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $-0.321 + 0.946i$
Analytic conductor: \(38.2323\)
Root analytic conductor: \(6.18323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4788} (1261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4788,\ (\ :1/2),\ -0.321 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.444039429\)
\(L(\frac12)\) \(\approx\) \(1.444039429\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
19 \( 1 + (0.5 - 4.33i)T \)
good5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 11T + 37T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6 + 10.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.158298450776407559672973001040, −7.78639994038734973307461678965, −6.55651818616709646887846541346, −5.67067376810769761117609707271, −5.16951566271558262363643741375, −4.32949940606429378621455867667, −3.66359161650387176631355404037, −2.61518718276400423931969618664, −1.38441409092344881405235100626, −0.47441046482000294522456104547, 1.06408567950605499500584299816, 2.70067309357954853031088170099, 2.77774129523167797632119748128, 4.16314529839122905723002712160, 4.55352544663679316323588581710, 5.71919504845795511469764264071, 6.39095450806537816325683679262, 7.27916376875473359237491123971, 7.54650909425415982497465392120, 8.399542286325577962819557474935

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