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Properties

Label 2-1740-1740.119-c0-0-1
Degree $2$
Conductor $1740$
Sign $0.613 + 0.789i$
Analytic cond. $0.868373$
Root an. cond. $0.931865$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.532 − 0.846i)2-s + (0.993 − 0.111i)3-s + (−0.433 − 0.900i)4-s + (0.222 + 0.974i)5-s + (0.433 − 0.900i)6-s + (0.595 + 0.286i)7-s + (−0.993 − 0.111i)8-s + (0.974 − 0.222i)9-s + (0.943 + 0.330i)10-s + (−0.532 − 0.846i)12-s + (0.559 − 0.351i)14-s + (0.330 + 0.943i)15-s + (−0.623 + 0.781i)16-s + (0.330 − 0.943i)18-s + (0.781 − 0.623i)20-s + (0.623 + 0.218i)21-s + ⋯
L(s)  = 1  + (0.532 − 0.846i)2-s + (0.993 − 0.111i)3-s + (−0.433 − 0.900i)4-s + (0.222 + 0.974i)5-s + (0.433 − 0.900i)6-s + (0.595 + 0.286i)7-s + (−0.993 − 0.111i)8-s + (0.974 − 0.222i)9-s + (0.943 + 0.330i)10-s + (−0.532 − 0.846i)12-s + (0.559 − 0.351i)14-s + (0.330 + 0.943i)15-s + (−0.623 + 0.781i)16-s + (0.330 − 0.943i)18-s + (0.781 − 0.623i)20-s + (0.623 + 0.218i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $0.613 + 0.789i$
Analytic conductor: \(0.868373\)
Root analytic conductor: \(0.931865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1740} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1740,\ (\ :0),\ 0.613 + 0.789i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.096007623\)
\(L(\frac12)\) \(\approx\) \(2.096007623\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.532 + 0.846i)T \)
3 \( 1 + (-0.993 + 0.111i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
good7 \( 1 + (-0.595 - 0.286i)T + (0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.974 + 0.222i)T^{2} \)
13 \( 1 + (-0.222 - 0.974i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (0.781 + 0.623i)T^{2} \)
23 \( 1 + (1.65 + 0.376i)T + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (-0.433 - 0.900i)T^{2} \)
37 \( 1 + (0.974 + 0.222i)T^{2} \)
41 \( 1 + (0.158 - 0.158i)T - iT^{2} \)
43 \( 1 + (0.663 + 1.05i)T + (-0.433 + 0.900i)T^{2} \)
47 \( 1 + (0.0971 + 0.862i)T + (-0.974 + 0.222i)T^{2} \)
53 \( 1 + (-0.900 + 0.433i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (1.33 - 0.467i)T + (0.781 - 0.623i)T^{2} \)
67 \( 1 + (-1.10 + 0.881i)T + (0.222 - 0.974i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.433 + 0.900i)T^{2} \)
79 \( 1 + (-0.974 - 0.222i)T^{2} \)
83 \( 1 + (0.862 + 1.79i)T + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (1.00 - 1.59i)T + (-0.433 - 0.900i)T^{2} \)
97 \( 1 + (-0.781 - 0.623i)T^{2} \)
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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

−9.602487677370379377657836927747, −8.651402524167986793157458458025, −8.005833525614887619706019264082, −6.92230764610366839791747995588, −6.17749522888348965461900195901, −5.11105369148976515338808855935, −4.09817315736414361565262204626, −3.32225190361947949223335008935, −2.40934874055257219957072311253, −1.72755683829310083514998958644, 1.63958345341123962428781799130, 2.91612436592960410713030223572, 4.13516498236376735028673224285, 4.50828013283647984939192016867, 5.48560061363057781471100176774, 6.40430533847403254784558816639, 7.44946563701896806157106704151, 8.212793082354598036311940208442, 8.389650809903988190245516612878, 9.497009456831723252601365584738

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