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Properties

Label 2-1740-1740.1199-c0-0-0
Degree $2$
Conductor $1740$
Sign $-0.895 + 0.444i$
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 + (−0.846 − 0.532i)3-s + (−0.433 + 0.900i)4-s + (−0.222 + 0.974i)5-s − i·6-s + (−0.595 + 0.286i)7-s + (−0.993 + 0.111i)8-s + (0.433 + 0.900i)9-s + (−0.943 + 0.330i)10-s + (0.846 − 0.532i)12-s + (−0.559 − 0.351i)14-s + (0.707 − 0.707i)15-s + (−0.623 − 0.781i)16-s + (−0.532 + 0.846i)18-s + (−0.781 − 0.623i)20-s + (0.656 + 0.0739i)21-s + ⋯
L(s)  = 1  + (0.532 + 0.846i)2-s + (−0.846 − 0.532i)3-s + (−0.433 + 0.900i)4-s + (−0.222 + 0.974i)5-s − i·6-s + (−0.595 + 0.286i)7-s + (−0.993 + 0.111i)8-s + (0.433 + 0.900i)9-s + (−0.943 + 0.330i)10-s + (0.846 − 0.532i)12-s + (−0.559 − 0.351i)14-s + (0.707 − 0.707i)15-s + (−0.623 − 0.781i)16-s + (−0.532 + 0.846i)18-s + (−0.781 − 0.623i)20-s + (0.656 + 0.0739i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(0.868373\)
Root analytic conductor: \(0.931865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1740} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1740,\ (\ :0),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4691111561\)
\(L(\frac12)\) \(\approx\) \(0.4691111561\)
\(L(1)\) 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 \( 1 + (0.846 + 0.532i)T \)
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

−9.962335104669977269207511795885, −9.082470597713893878306599352852, −7.80943608954817649801741301221, −7.54669748496685851146490312874, −6.52884905551170815758485714190, −6.14368656727865957950755532882, −5.40291087833208038146624361486, −4.26585142495085827923821128241, −3.36873977710693800400460970707, −2.22174066818339001989796204113, 0.32041085711524132258861979827, 1.74389766448640885259457827645, 3.31870298196665659203483510600, 4.18080008049729548640368799035, 4.65841682806612252048419873245, 5.76884067590491060110861523267, 6.11113575832152632033329561038, 7.38563737077622741807297536707, 8.593353707639087477309795208906, 9.370498472602403710285334849529

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