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Properties

Label 2-1740-145.17-c1-0-14
Degree $2$
Conductor $1740$
Sign $0.805 + 0.592i$
Analytic cond. $13.8939$
Root an. cond. $3.72746$
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·3-s + (1.84 + 1.25i)5-s + (−2.83 − 2.83i)7-s − 9-s + (1.19 − 1.19i)11-s + (3.93 + 3.93i)13-s + (1.25 − 1.84i)15-s + 2.62·17-s + (2.85 + 2.85i)19-s + (−2.83 + 2.83i)21-s + (−1.35 + 1.35i)23-s + (1.83 + 4.65i)25-s + i·27-s + (−2.09 + 4.96i)29-s + (7.14 − 7.14i)31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.826 + 0.562i)5-s + (−1.06 − 1.06i)7-s − 0.333·9-s + (0.359 − 0.359i)11-s + (1.09 + 1.09i)13-s + (0.324 − 0.477i)15-s + 0.635·17-s + (0.654 + 0.654i)19-s + (−0.617 + 0.617i)21-s + (−0.283 + 0.283i)23-s + (0.367 + 0.930i)25-s + 0.192i·27-s + (−0.389 + 0.921i)29-s + (1.28 − 1.28i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $0.805 + 0.592i$
Analytic conductor: \(13.8939\)
Root analytic conductor: \(3.72746\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1740} (1177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1740,\ (\ :1/2),\ 0.805 + 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.939208890\)
\(L(\frac12)\) \(\approx\) \(1.939208890\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 + (-1.84 - 1.25i)T \)
29 \( 1 + (2.09 - 4.96i)T \)
good7 \( 1 + (2.83 + 2.83i)T + 7iT^{2} \)
11 \( 1 + (-1.19 + 1.19i)T - 11iT^{2} \)
13 \( 1 + (-3.93 - 3.93i)T + 13iT^{2} \)
17 \( 1 - 2.62T + 17T^{2} \)
19 \( 1 + (-2.85 - 2.85i)T + 19iT^{2} \)
23 \( 1 + (1.35 - 1.35i)T - 23iT^{2} \)
31 \( 1 + (-7.14 + 7.14i)T - 31iT^{2} \)
37 \( 1 + 7.00iT - 37T^{2} \)
41 \( 1 + (2.40 + 2.40i)T + 41iT^{2} \)
43 \( 1 + 1.12iT - 43T^{2} \)
47 \( 1 + 2.44iT - 47T^{2} \)
53 \( 1 + (-6.22 + 6.22i)T - 53iT^{2} \)
59 \( 1 + 8.02iT - 59T^{2} \)
61 \( 1 + (-4.31 + 4.31i)T - 61iT^{2} \)
67 \( 1 + (-8.36 + 8.36i)T - 67iT^{2} \)
71 \( 1 - 5.01iT - 71T^{2} \)
73 \( 1 + 6.52T + 73T^{2} \)
79 \( 1 + (-5.73 - 5.73i)T + 79iT^{2} \)
83 \( 1 + (-0.739 + 0.739i)T - 83iT^{2} \)
89 \( 1 + (-10.7 - 10.7i)T + 89iT^{2} \)
97 \( 1 + 12.8iT - 97T^{2} \)
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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

−9.466274202646046167454633910873, −8.431715187807581446054275251696, −7.42372747986273346417667028969, −6.72990288685140450972775683356, −6.25294395612334781127532199924, −5.46593596533217416453268319620, −3.83629638904642938698139528368, −3.41856400485244345635744282521, −2.04279996339614450827511560157, −0.942671110080761157782123095101, 1.07456236643836689001358051971, 2.63467125083859635960764778162, 3.28286244079132230660908364417, 4.53532644734307308778367385990, 5.50808518244409757214086517205, 5.95864670329563061291002310501, 6.73050522416843160600695722875, 8.153314426629486612308582352435, 8.744614138968144140996221654503, 9.459519413346012795687214014314

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