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Properties

Label 2-1740-1740.347-c0-0-5
Degree $2$
Conductor $1740$
Sign $-0.229 + 0.973i$
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  + (0.707 − 0.707i)2-s + (−0.965 − 0.258i)3-s − 1.00i·4-s + (−0.5 + 0.866i)5-s + (−0.866 + 0.500i)6-s + (−0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (0.258 + 0.965i)10-s + 0.517i·11-s + (−0.258 + 0.965i)12-s + (1.36 − 1.36i)13-s + (0.707 − 0.707i)15-s − 1.00·16-s + (0.965 − 0.258i)18-s − 1.41i·19-s + (0.866 + 0.500i)20-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.965 − 0.258i)3-s − 1.00i·4-s + (−0.5 + 0.866i)5-s + (−0.866 + 0.500i)6-s + (−0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (0.258 + 0.965i)10-s + 0.517i·11-s + (−0.258 + 0.965i)12-s + (1.36 − 1.36i)13-s + (0.707 − 0.707i)15-s − 1.00·16-s + (0.965 − 0.258i)18-s − 1.41i·19-s + (0.866 + 0.500i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(0.868373\)
Root analytic conductor: \(0.931865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1740} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1740,\ (\ :0),\ -0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.086783842\)
\(L(\frac12)\) \(\approx\) \(1.086783842\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 - T \)
good7 \( 1 - iT^{2} \)
11 \( 1 - 0.517iT - T^{2} \)
13 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 - iT^{2} \)
31 \( 1 + 0.517T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
47 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
53 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.93iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.657796979571782503613919031819, −8.473392860679795706226785812103, −7.38755849148147678898825993087, −6.69081341996871401206159788725, −5.97335040701597973986289136219, −5.18014681373949497653088573181, −4.25327111755079999250565386943, −3.35886995995377478317577523114, −2.34291945855989740882367145793, −0.838503811596861825361560945665, 1.43586068561049248695915439248, 3.47244407685403312796160907920, 4.18493898588739507400545474907, 4.74723220978696875455658568045, 5.95474549393931669110794014553, 6.08780198951945544420019963803, 7.20513471746200032764615381903, 8.073952656507309344940773190610, 8.815018724242923303950302729264, 9.497788696431433925358363917421

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