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Properties

Label 2-3330-37.36-c1-0-47
Degree $2$
Conductor $3330$
Sign $0.806 + 0.590i$
Analytic cond. $26.5901$
Root an. cond. $5.15656$
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  + i·2-s − 4-s + i·5-s + 2.75·7-s − i·8-s − 10-s − 4.75·11-s + 6.75i·13-s + 2.75i·14-s + 16-s − 7.06i·17-s − 6.75i·19-s − i·20-s − 4.75i·22-s − 5.06i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.447i·5-s + 1.03·7-s − 0.353i·8-s − 0.316·10-s − 1.43·11-s + 1.87i·13-s + 0.735i·14-s + 0.250·16-s − 1.71i·17-s − 1.54i·19-s − 0.223i·20-s − 1.01i·22-s − 1.05i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $0.806 + 0.590i$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3330} (2071, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ 0.806 + 0.590i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.130049118\)
\(L(\frac12)\) \(\approx\) \(1.130049118\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 - iT \)
37 \( 1 + (4.90 + 3.59i)T \)
good7 \( 1 - 2.75T + 7T^{2} \)
11 \( 1 + 4.75T + 11T^{2} \)
13 \( 1 - 6.75iT - 13T^{2} \)
17 \( 1 + 7.06iT - 17T^{2} \)
19 \( 1 + 6.75iT - 19T^{2} \)
23 \( 1 + 5.06iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 0.315iT - 31T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 9.81iT - 43T^{2} \)
47 \( 1 + 3.50T + 47T^{2} \)
53 \( 1 + 4.75T + 53T^{2} \)
59 \( 1 - 11.8iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 - 15.0T + 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 + 6.43T + 73T^{2} \)
79 \( 1 + 8.31iT - 79T^{2} \)
83 \( 1 + 4.43T + 83T^{2} \)
89 \( 1 + 6.25iT - 89T^{2} \)
97 \( 1 + 11.5iT - 97T^{2} \)
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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

−8.566350234328246034271097515544, −7.59464253810657816849378016858, −7.11490054215402817911564506022, −6.55461846839316557573632389781, −5.32946103628406630970720755729, −4.86906173607985333338579141798, −4.21594424297062005361230934068, −2.79068267126582215474939434480, −2.07927365534876293991610865485, −0.35618708706655334385026727382, 1.20115089309241207821207885068, 1.98851913494677918516264586813, 3.17170236955181322779549761860, 3.85537860899494952358593414633, 5.11677433029320057041542865025, 5.29218376590276322630207137104, 6.18158977351152143742208782553, 7.81529707268305564567150710005, 8.069221660996721451417729563258, 8.336377778688631919024575615120

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