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Properties

Label 2-1755-9.4-c1-0-34
Degree $2$
Conductor $1755$
Sign $0.708 + 0.705i$
Analytic cond. $14.0137$
Root an. cond. $3.74349$
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  + (0.327 + 0.567i)2-s + (0.785 − 1.36i)4-s + (−0.5 + 0.866i)5-s + (0.388 + 0.673i)7-s + 2.33·8-s − 0.655·10-s + (−2.03 − 3.52i)11-s + (0.5 − 0.866i)13-s + (−0.254 + 0.440i)14-s + (−0.804 − 1.39i)16-s − 5.29·17-s + 6.65·19-s + (0.785 + 1.36i)20-s + (1.33 − 2.30i)22-s + (1.15 − 1.99i)23-s + ⋯
L(s)  = 1  + (0.231 + 0.401i)2-s + (0.392 − 0.680i)4-s + (−0.223 + 0.387i)5-s + (0.146 + 0.254i)7-s + 0.827·8-s − 0.207·10-s + (−0.613 − 1.06i)11-s + (0.138 − 0.240i)13-s + (−0.0680 + 0.117i)14-s + (−0.201 − 0.348i)16-s − 1.28·17-s + 1.52·19-s + (0.175 + 0.304i)20-s + (0.284 − 0.492i)22-s + (0.240 − 0.416i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $0.708 + 0.705i$
Analytic conductor: \(14.0137\)
Root analytic conductor: \(3.74349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1755} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1755,\ (\ :1/2),\ 0.708 + 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.996735132\)
\(L(\frac12)\) \(\approx\) \(1.996735132\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.327 - 0.567i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-0.388 - 0.673i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.03 + 3.52i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
19 \( 1 - 6.65T + 19T^{2} \)
23 \( 1 + (-1.15 + 1.99i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.19 - 5.52i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.47 + 7.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.80T + 37T^{2} \)
41 \( 1 + (-3.80 + 6.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.42 + 7.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.04 + 3.54i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.78T + 53T^{2} \)
59 \( 1 + (2.37 - 4.10i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.18 - 12.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.39 - 9.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.307T + 71T^{2} \)
73 \( 1 + 7.50T + 73T^{2} \)
79 \( 1 + (3.80 + 6.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.26 + 10.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 + (-3.04 - 5.27i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.046207504924877896462065202216, −8.415351599779124884782798175782, −7.38103143808476255021355193355, −6.86495669661531356181631899427, −5.84269861313193176138695545272, −5.39823259789680562731959021574, −4.37844028037083008328173716845, −3.14884512718709174607199873632, −2.23308968658191688819576641188, −0.71982040254904999201101255395, 1.39064104799783918249571130095, 2.50868800966064362767203901691, 3.40477552535986524026567642651, 4.57262452990260698784289902218, 4.85183423814257189832107921154, 6.34740365423199331304496941172, 7.17141059762002306236549840107, 7.77450070464527641131055173428, 8.485688422397499215926523691804, 9.496526216224996877271438662909

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