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Properties

Label 2-1755-9.4-c1-0-6
Degree $2$
Conductor $1755$
Sign $0.915 - 0.402i$
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  + (−1.29 − 2.23i)2-s + (−2.34 + 4.05i)4-s + (−0.5 + 0.866i)5-s + (2.13 + 3.70i)7-s + 6.95·8-s + 2.58·10-s + (2.19 + 3.79i)11-s + (0.5 − 0.866i)13-s + (5.52 − 9.57i)14-s + (−4.30 − 7.44i)16-s − 0.619·17-s − 2.61·19-s + (−2.34 − 4.05i)20-s + (5.66 − 9.81i)22-s + (2.71 − 4.70i)23-s + ⋯
L(s)  = 1  + (−0.914 − 1.58i)2-s + (−1.17 + 2.02i)4-s + (−0.223 + 0.387i)5-s + (0.807 + 1.39i)7-s + 2.45·8-s + 0.817·10-s + (0.660 + 1.14i)11-s + (0.138 − 0.240i)13-s + (1.47 − 2.55i)14-s + (−1.07 − 1.86i)16-s − 0.150·17-s − 0.599·19-s + (−0.524 − 0.907i)20-s + (1.20 − 2.09i)22-s + (0.565 − 0.980i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $0.915 - 0.402i$
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: \((2,\ 1755,\ (\ :1/2),\ 0.915 - 0.402i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9166314023\)
\(L(\frac12)\) \(\approx\) \(0.9166314023\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.29 + 2.23i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-2.13 - 3.70i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.19 - 3.79i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.619T + 17T^{2} \)
19 \( 1 + 2.61T + 19T^{2} \)
23 \( 1 + (-2.71 + 4.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.47 - 6.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.44 + 5.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.02T + 37T^{2} \)
41 \( 1 + (2.03 - 3.52i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.01 - 8.69i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.202 + 0.351i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.71T + 53T^{2} \)
59 \( 1 + (5.36 - 9.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.53 + 4.39i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.27 - 12.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.57T + 71T^{2} \)
73 \( 1 - 16.7T + 73T^{2} \)
79 \( 1 + (7.27 + 12.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.52 + 2.63i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.27T + 89T^{2} \)
97 \( 1 + (-4.81 - 8.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.347919389315996819231553938511, −8.777984255256938382716402337247, −8.236093002309386489697253170881, −7.31762051320875149374219062145, −6.22653003940945590384808736749, −4.84705954883654766700951214447, −4.19145744067506768367537223964, −2.89587729342422120689252919661, −2.29770196076229465605292339693, −1.33648259039146704898711539456, 0.55218720073145731570811609969, 1.41779840978845218530491075776, 3.72235287516645626711741711078, 4.55326957473091231991413610463, 5.38742664600955712892710789767, 6.32514983827735714273215959692, 7.02318795821476935314239168226, 7.65839142630137435923634173022, 8.477782906127169004085006443172, 8.788906615422209710237976376283

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