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Properties

Label 2-1740-1740.1043-c0-0-0
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 + (0.707 + 0.707i)3-s + 1.00i·4-s − 5-s − 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (0.707 + 0.707i)10-s + 1.41i·11-s + (−0.707 + 0.707i)12-s + (−1 − i)13-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (0.707 − 0.707i)18-s + 1.41i·19-s − 1.00i·20-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s − 5-s − 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (0.707 + 0.707i)10-s + 1.41i·11-s + (−0.707 + 0.707i)12-s + (−1 − i)13-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (0.707 − 0.707i)18-s + 1.41i·19-s − 1.00i·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: $\chi_{1740} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1740,\ (\ :0),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5658682377\)
\(L(\frac12)\) \(\approx\) \(0.5658682377\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + T \)
29 \( 1 + T \)
good7 \( 1 + iT^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 + iT^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
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

−9.673586146709854814419148082384, −9.164337390506796476936687782678, −8.116530669330064201401799829240, −7.64826527279626369470986786456, −7.18046212213850842662167809254, −5.36580711791669386197330933394, −4.38664229123118070110858039993, −3.77057792233617898319802442374, −2.86313692905934439745797695167, −1.84903312700572426965149304934, 0.48129827162929123632247103340, 2.02213743222739521989328895069, 3.19982552445945325989841952820, 4.29558010098397169672607966131, 5.39764602599749988310517202423, 6.44743638520260574001519333281, 7.22517008712926688547946311891, 7.54526373177851975103665497719, 8.570831590406077037419440164144, 8.937227571199400296905991848410

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