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Properties

Label 2-1740-145.17-c1-0-27
Degree $2$
Conductor $1740$
Sign $-0.910 + 0.414i$
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 + (2.23 − 0.128i)5-s + (−1.28 − 1.28i)7-s − 9-s + (1.05 − 1.05i)11-s + (−2.22 − 2.22i)13-s + (−0.128 − 2.23i)15-s − 2.52·17-s + (−4.87 − 4.87i)19-s + (−1.28 + 1.28i)21-s + (−5.59 + 5.59i)23-s + (4.96 − 0.575i)25-s + i·27-s + (−0.0348 − 5.38i)29-s + (−2.89 + 2.89i)31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.998 − 0.0576i)5-s + (−0.485 − 0.485i)7-s − 0.333·9-s + (0.318 − 0.318i)11-s + (−0.616 − 0.616i)13-s + (−0.0332 − 0.576i)15-s − 0.612·17-s + (−1.11 − 1.11i)19-s + (−0.280 + 0.280i)21-s + (−1.16 + 1.16i)23-s + (0.993 − 0.115i)25-s + 0.192i·27-s + (−0.00647 − 0.999i)29-s + (−0.519 + 0.519i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $-0.910 + 0.414i$
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: \((2,\ 1740,\ (\ :1/2),\ -0.910 + 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.087078602\)
\(L(\frac12)\) \(\approx\) \(1.087078602\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 0.128i)T \)
29 \( 1 + (0.0348 + 5.38i)T \)
good7 \( 1 + (1.28 + 1.28i)T + 7iT^{2} \)
11 \( 1 + (-1.05 + 1.05i)T - 11iT^{2} \)
13 \( 1 + (2.22 + 2.22i)T + 13iT^{2} \)
17 \( 1 + 2.52T + 17T^{2} \)
19 \( 1 + (4.87 + 4.87i)T + 19iT^{2} \)
23 \( 1 + (5.59 - 5.59i)T - 23iT^{2} \)
31 \( 1 + (2.89 - 2.89i)T - 31iT^{2} \)
37 \( 1 + 1.36iT - 37T^{2} \)
41 \( 1 + (0.246 + 0.246i)T + 41iT^{2} \)
43 \( 1 + 5.09iT - 43T^{2} \)
47 \( 1 - 12.7iT - 47T^{2} \)
53 \( 1 + (-4.71 + 4.71i)T - 53iT^{2} \)
59 \( 1 + 10.1iT - 59T^{2} \)
61 \( 1 + (-2.42 + 2.42i)T - 61iT^{2} \)
67 \( 1 + (6.93 - 6.93i)T - 67iT^{2} \)
71 \( 1 + 3.88iT - 71T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 + (4.75 + 4.75i)T + 79iT^{2} \)
83 \( 1 + (-2.68 + 2.68i)T - 83iT^{2} \)
89 \( 1 + (1.15 + 1.15i)T + 89iT^{2} \)
97 \( 1 - 10.0iT - 97T^{2} \)
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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

−9.059689559610692622291217260100, −8.179461724130886117336556591692, −7.25166648813898152545779398738, −6.53941311101130369101559989403, −5.91326296239964457604090612430, −4.99723451587854518105394865877, −3.86551172551966460371706806244, −2.68579330495364733726271733975, −1.83572904045049131029507396583, −0.36780185882411908530353646733, 1.87885031709831223847426114869, 2.59450856243430616346179349393, 3.92118359916463915651045940103, 4.67602629763048141061164039283, 5.73463644720864513936336338561, 6.31102128759468695614800043344, 7.06444293668554016163707587685, 8.392693336149782246683743972332, 8.972467791643941904631732362594, 9.697963915595391528341133522678

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