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Properties

Label 2-1740-145.128-c1-0-26
Degree $2$
Conductor $1740$
Sign $-0.663 + 0.747i$
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 + (−2.22 + 0.226i)5-s + (2.93 − 2.93i)7-s − 9-s + (−0.699 − 0.699i)11-s + (1.95 − 1.95i)13-s + (−0.226 − 2.22i)15-s − 3.39·17-s + (−4.54 + 4.54i)19-s + (2.93 + 2.93i)21-s + (0.532 + 0.532i)23-s + (4.89 − 1.00i)25-s − i·27-s + (−4.81 − 2.41i)29-s + (−5.95 − 5.95i)31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.994 + 0.101i)5-s + (1.10 − 1.10i)7-s − 0.333·9-s + (−0.211 − 0.211i)11-s + (0.541 − 0.541i)13-s + (−0.0585 − 0.574i)15-s − 0.823·17-s + (−1.04 + 1.04i)19-s + (0.640 + 0.640i)21-s + (0.110 + 0.110i)23-s + (0.979 − 0.201i)25-s − 0.192i·27-s + (−0.894 − 0.447i)29-s + (−1.06 − 1.06i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 + 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $-0.663 + 0.747i$
Analytic conductor: \(13.8939\)
Root analytic conductor: \(3.72746\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1740} (853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1740,\ (\ :1/2),\ -0.663 + 0.747i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5159909326\)
\(L(\frac12)\) \(\approx\) \(0.5159909326\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 - 0.226i)T \)
29 \( 1 + (4.81 + 2.41i)T \)
good7 \( 1 + (-2.93 + 2.93i)T - 7iT^{2} \)
11 \( 1 + (0.699 + 0.699i)T + 11iT^{2} \)
13 \( 1 + (-1.95 + 1.95i)T - 13iT^{2} \)
17 \( 1 + 3.39T + 17T^{2} \)
19 \( 1 + (4.54 - 4.54i)T - 19iT^{2} \)
23 \( 1 + (-0.532 - 0.532i)T + 23iT^{2} \)
31 \( 1 + (5.95 + 5.95i)T + 31iT^{2} \)
37 \( 1 + 5.03iT - 37T^{2} \)
41 \( 1 + (7.88 - 7.88i)T - 41iT^{2} \)
43 \( 1 - 8.53iT - 43T^{2} \)
47 \( 1 + 0.972iT - 47T^{2} \)
53 \( 1 + (7.28 + 7.28i)T + 53iT^{2} \)
59 \( 1 - 0.720iT - 59T^{2} \)
61 \( 1 + (1.46 + 1.46i)T + 61iT^{2} \)
67 \( 1 + (-9.47 - 9.47i)T + 67iT^{2} \)
71 \( 1 + 9.49iT - 71T^{2} \)
73 \( 1 + 7.20T + 73T^{2} \)
79 \( 1 + (-7.50 + 7.50i)T - 79iT^{2} \)
83 \( 1 + (11.6 + 11.6i)T + 83iT^{2} \)
89 \( 1 + (-0.482 + 0.482i)T - 89iT^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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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

−8.869118236956265694927659781607, −8.010450803836423982959056072991, −7.81034006738580653302145866958, −6.71339016675858298680493043944, −5.67488473208709916771827494922, −4.59273568107074480554159761013, −4.09425257025257746430723231535, −3.33110771058708813588083735422, −1.75479759633027197644322431818, −0.18930267590757807427784947079, 1.61517341370511026612619624080, 2.48109706707237339957766997194, 3.77284679246062099933617206495, 4.78281683416048597502023084518, 5.39724966689820143787457448209, 6.64398307654113388918319454559, 7.17621639166830195993933067948, 8.201903619072628484296400122145, 8.698685246719301660954447353452, 9.123744165789317457974836965180

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