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Properties

Label 2-1740-145.128-c1-0-24
Degree $2$
Conductor $1740$
Sign $-0.0462 + 0.998i$
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 + (0.304 + 2.21i)5-s + (−0.0397 + 0.0397i)7-s − 9-s + (−3.24 − 3.24i)11-s + (3.48 − 3.48i)13-s + (−2.21 + 0.304i)15-s − 6.30·17-s + (1.53 − 1.53i)19-s + (−0.0397 − 0.0397i)21-s + (−4.76 − 4.76i)23-s + (−4.81 + 1.34i)25-s − i·27-s + (−0.967 − 5.29i)29-s + (−2.66 − 2.66i)31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.135 + 0.990i)5-s + (−0.0150 + 0.0150i)7-s − 0.333·9-s + (−0.977 − 0.977i)11-s + (0.965 − 0.965i)13-s + (−0.571 + 0.0784i)15-s − 1.53·17-s + (0.351 − 0.351i)19-s + (−0.00866 − 0.00866i)21-s + (−0.992 − 0.992i)23-s + (−0.963 + 0.269i)25-s − 0.192i·27-s + (−0.179 − 0.983i)29-s + (−0.479 − 0.479i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0462 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0462 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $-0.0462 + 0.998i$
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: \((2,\ 1740,\ (\ :1/2),\ -0.0462 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6307635123\)
\(L(\frac12)\) \(\approx\) \(0.6307635123\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.304 - 2.21i)T \)
29 \( 1 + (0.967 + 5.29i)T \)
good7 \( 1 + (0.0397 - 0.0397i)T - 7iT^{2} \)
11 \( 1 + (3.24 + 3.24i)T + 11iT^{2} \)
13 \( 1 + (-3.48 + 3.48i)T - 13iT^{2} \)
17 \( 1 + 6.30T + 17T^{2} \)
19 \( 1 + (-1.53 + 1.53i)T - 19iT^{2} \)
23 \( 1 + (4.76 + 4.76i)T + 23iT^{2} \)
31 \( 1 + (2.66 + 2.66i)T + 31iT^{2} \)
37 \( 1 - 7.72iT - 37T^{2} \)
41 \( 1 + (-1.73 + 1.73i)T - 41iT^{2} \)
43 \( 1 - 0.104iT - 43T^{2} \)
47 \( 1 + 2.24iT - 47T^{2} \)
53 \( 1 + (5.73 + 5.73i)T + 53iT^{2} \)
59 \( 1 - 0.752iT - 59T^{2} \)
61 \( 1 + (0.412 + 0.412i)T + 61iT^{2} \)
67 \( 1 + (9.66 + 9.66i)T + 67iT^{2} \)
71 \( 1 + 2.28iT - 71T^{2} \)
73 \( 1 - 6.70T + 73T^{2} \)
79 \( 1 + (0.593 - 0.593i)T - 79iT^{2} \)
83 \( 1 + (-8.73 - 8.73i)T + 83iT^{2} \)
89 \( 1 + (-9.03 + 9.03i)T - 89iT^{2} \)
97 \( 1 - 16.6iT - 97T^{2} \)
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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

−9.136611876321501537304601986633, −8.250833438224360899014457129706, −7.74089803721255641438546719977, −6.39665950865319325878589364528, −6.06698460542363618332445401405, −5.03734702064872989858869529316, −3.94956279318970710087755898778, −3.08915146318343436936926196371, −2.31720577442188966337800633680, −0.22289431829736300590717079850, 1.52216793019635377849328416901, 2.19168621615595100259266128806, 3.75766610814242522989648251788, 4.62374941301343756164859673403, 5.46942075182278472193087140292, 6.30290547763785709411732009255, 7.24284069990762230724176489028, 7.889326593331979889677261722760, 8.859047934297103758871386870033, 9.221485122895508763643745181214

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