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Properties

Label 2-1740-145.128-c1-0-5
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: $\chi_{1740} (853, \cdot )$
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

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

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