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Properties

Label 2-1755-65.38-c0-0-0
Degree $2$
Conductor $1755$
Sign $0.774 + 0.632i$
Analytic cond. $0.875859$
Root an. cond. $0.935873$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.40 − 1.40i)2-s + 2.93i·4-s + (−0.991 − 0.130i)5-s + (2.70 − 2.70i)8-s + (1.20 + 1.57i)10-s + 0.765i·11-s + (−0.707 − 0.707i)13-s − 4.66·16-s + (0.382 − 2.90i)20-s + (1.07 − 1.07i)22-s + (0.965 + 0.258i)25-s + 1.98i·26-s + (3.83 + 3.83i)32-s + (−3.03 + 2.33i)40-s + 1.84i·41-s + ⋯
L(s)  = 1  + (−1.40 − 1.40i)2-s + 2.93i·4-s + (−0.991 − 0.130i)5-s + (2.70 − 2.70i)8-s + (1.20 + 1.57i)10-s + 0.765i·11-s + (−0.707 − 0.707i)13-s − 4.66·16-s + (0.382 − 2.90i)20-s + (1.07 − 1.07i)22-s + (0.965 + 0.258i)25-s + 1.98i·26-s + (3.83 + 3.83i)32-s + (−3.03 + 2.33i)40-s + 1.84i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $0.774 + 0.632i$
Analytic conductor: \(0.875859\)
Root analytic conductor: \(0.935873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1755} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1755,\ (\ :0),\ 0.774 + 0.632i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3883832474\)
\(L(\frac12)\) \(\approx\) \(0.3883832474\)
\(L(1)\) 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 \( 1 + (0.991 + 0.130i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (1.40 + 1.40i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 - 0.765iT - T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - 1.84iT - T^{2} \)
43 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
47 \( 1 + (-0.184 - 0.184i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - 1.58T + T^{2} \)
61 \( 1 - 0.517T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 0.261iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + (-1.12 + 1.12i)T - iT^{2} \)
89 \( 1 - 1.21T + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.582332210457383828167857556631, −8.737555175966307394701406702327, −7.956976309670575251491976404711, −7.56997310438101791037132617023, −6.73811137015250040136090468618, −4.86457873615204974109143090436, −4.08287100298377337396250742174, −3.15130176452380998613306058525, −2.31201525716207892213604577366, −0.942323973296833461071814327417, 0.64081702636575148810204520186, 2.24047105494698969823889648048, 3.97928980116149340544942059610, 5.01304497425822908117196152538, 5.82728072170197257087923547400, 6.76274720081772384960599628276, 7.32129694838708511747442665198, 7.920787669813350972863363357377, 8.801921291596921572223055502998, 9.115624191805193425848267560529

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