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Properties

Label 2-1740-145.17-c1-0-28
Degree $2$
Conductor $1740$
Sign $-0.395 - 0.918i$
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 + (−2.01 − 0.965i)5-s + (−2.62 − 2.62i)7-s − 9-s + (2.92 − 2.92i)11-s + (−0.913 − 0.913i)13-s + (−0.965 + 2.01i)15-s − 7.51·17-s + (0.797 + 0.797i)19-s + (−2.62 + 2.62i)21-s + (2.25 − 2.25i)23-s + (3.13 + 3.89i)25-s + i·27-s + (−4.75 + 2.53i)29-s + (2.09 − 2.09i)31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.901 − 0.431i)5-s + (−0.992 − 0.992i)7-s − 0.333·9-s + (0.883 − 0.883i)11-s + (−0.253 − 0.253i)13-s + (−0.249 + 0.520i)15-s − 1.82·17-s + (0.182 + 0.182i)19-s + (−0.572 + 0.572i)21-s + (0.470 − 0.470i)23-s + (0.627 + 0.778i)25-s + 0.192i·27-s + (−0.882 + 0.469i)29-s + (0.375 − 0.375i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $-0.395 - 0.918i$
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: \((2,\ 1740,\ (\ :1/2),\ -0.395 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2215086394\)
\(L(\frac12)\) \(\approx\) \(0.2215086394\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.01 + 0.965i)T \)
29 \( 1 + (4.75 - 2.53i)T \)
good7 \( 1 + (2.62 + 2.62i)T + 7iT^{2} \)
11 \( 1 + (-2.92 + 2.92i)T - 11iT^{2} \)
13 \( 1 + (0.913 + 0.913i)T + 13iT^{2} \)
17 \( 1 + 7.51T + 17T^{2} \)
19 \( 1 + (-0.797 - 0.797i)T + 19iT^{2} \)
23 \( 1 + (-2.25 + 2.25i)T - 23iT^{2} \)
31 \( 1 + (-2.09 + 2.09i)T - 31iT^{2} \)
37 \( 1 + 0.0487iT - 37T^{2} \)
41 \( 1 + (-3.75 - 3.75i)T + 41iT^{2} \)
43 \( 1 - 0.0330iT - 43T^{2} \)
47 \( 1 + 2.62iT - 47T^{2} \)
53 \( 1 + (-6.50 + 6.50i)T - 53iT^{2} \)
59 \( 1 - 10.9iT - 59T^{2} \)
61 \( 1 + (-0.249 + 0.249i)T - 61iT^{2} \)
67 \( 1 + (9.90 - 9.90i)T - 67iT^{2} \)
71 \( 1 + 0.838iT - 71T^{2} \)
73 \( 1 - 3.36T + 73T^{2} \)
79 \( 1 + (-5.06 - 5.06i)T + 79iT^{2} \)
83 \( 1 + (10.1 - 10.1i)T - 83iT^{2} \)
89 \( 1 + (9.59 + 9.59i)T + 89iT^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
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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

−8.772070867711005912296002790566, −7.966144651036938544116189791178, −6.96255090293903808859758380155, −6.73068723691284212809855493054, −5.63218482187697208061944098187, −4.34258376860858921646919189797, −3.78084753786864487724513486133, −2.76078340935979603661869283189, −1.09309722451808203246965986989, −0.094593378215346608421253743674, 2.18534391582203803811464869572, 3.11410072956369870397383730344, 4.07130062887782382863923645108, 4.71169617327424648092240627616, 5.93717507668031176564947937898, 6.72680343709218099123737624455, 7.29897612896617232592364724719, 8.501062681222394262476968619986, 9.255915600376505778842636736391, 9.534834724704969694533574226710

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