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Properties

Label 2-1740-145.128-c1-0-21
Degree $2$
Conductor $1740$
Sign $0.740 + 0.672i$
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 + (2.12 − 0.680i)5-s + (2.56 − 2.56i)7-s − 9-s + (−2.95 − 2.95i)11-s + (−2.96 + 2.96i)13-s + (0.680 + 2.12i)15-s + 1.42·17-s + (4.21 − 4.21i)19-s + (2.56 + 2.56i)21-s + (−0.626 − 0.626i)23-s + (4.07 − 2.90i)25-s − i·27-s + (3.40 − 4.17i)29-s + (1.57 + 1.57i)31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.952 − 0.304i)5-s + (0.969 − 0.969i)7-s − 0.333·9-s + (−0.890 − 0.890i)11-s + (−0.822 + 0.822i)13-s + (0.175 + 0.549i)15-s + 0.346·17-s + (0.967 − 0.967i)19-s + (0.559 + 0.559i)21-s + (−0.130 − 0.130i)23-s + (0.814 − 0.580i)25-s − 0.192i·27-s + (0.631 − 0.775i)29-s + (0.282 + 0.282i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $0.740 + 0.672i$
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: \((2,\ 1740,\ (\ :1/2),\ 0.740 + 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.071965095\)
\(L(\frac12)\) \(\approx\) \(2.071965095\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 + 0.680i)T \)
29 \( 1 + (-3.40 + 4.17i)T \)
good7 \( 1 + (-2.56 + 2.56i)T - 7iT^{2} \)
11 \( 1 + (2.95 + 2.95i)T + 11iT^{2} \)
13 \( 1 + (2.96 - 2.96i)T - 13iT^{2} \)
17 \( 1 - 1.42T + 17T^{2} \)
19 \( 1 + (-4.21 + 4.21i)T - 19iT^{2} \)
23 \( 1 + (0.626 + 0.626i)T + 23iT^{2} \)
31 \( 1 + (-1.57 - 1.57i)T + 31iT^{2} \)
37 \( 1 + 3.65iT - 37T^{2} \)
41 \( 1 + (2.55 - 2.55i)T - 41iT^{2} \)
43 \( 1 - 6.27iT - 43T^{2} \)
47 \( 1 + 10.1iT - 47T^{2} \)
53 \( 1 + (-0.0167 - 0.0167i)T + 53iT^{2} \)
59 \( 1 + 1.42iT - 59T^{2} \)
61 \( 1 + (6.01 + 6.01i)T + 61iT^{2} \)
67 \( 1 + (-4.13 - 4.13i)T + 67iT^{2} \)
71 \( 1 - 5.04iT - 71T^{2} \)
73 \( 1 + 5.65T + 73T^{2} \)
79 \( 1 + (-9.12 + 9.12i)T - 79iT^{2} \)
83 \( 1 + (5.53 + 5.53i)T + 83iT^{2} \)
89 \( 1 + (-2.80 + 2.80i)T - 89iT^{2} \)
97 \( 1 - 1.50iT - 97T^{2} \)
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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

−9.335693232806894444144952961173, −8.476468110386911260478614414779, −7.73481962205920033940739340548, −6.85480574029121886964445422568, −5.80100910089456379355871141640, −4.94394622804513272838502792576, −4.55904418659389477797114609943, −3.21037749763743511674820821227, −2.16143858690181223893760968645, −0.806212315752611928183011438169, 1.45900689516105290653101827426, 2.32443595567461305516454851528, 3.05481791997683770340842541347, 4.88168060885841332262177257264, 5.35966815719310010481490037286, 6.02233564511117620155174878008, 7.19284634185898262429862462263, 7.77128370287678399781617364431, 8.479351602422674443457660680917, 9.497936929616407563029657560200

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