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Properties

Label 2-1755-65.38-c0-0-7
Degree $2$
Conductor $1755$
Sign $-0.632 + 0.774i$
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  + (−0.184 − 0.184i)2-s − 0.931i·4-s + (0.130 − 0.991i)5-s + (−0.356 + 0.356i)8-s + (−0.207 + 0.158i)10-s − 1.84i·11-s + (0.707 + 0.707i)13-s − 0.800·16-s + (−0.923 − 0.121i)20-s + (−0.341 + 0.341i)22-s + (−0.965 − 0.258i)25-s − 0.261i·26-s + (0.504 + 0.504i)32-s + (0.307 + 0.400i)40-s + 0.765i·41-s + ⋯
L(s)  = 1  + (−0.184 − 0.184i)2-s − 0.931i·4-s + (0.130 − 0.991i)5-s + (−0.356 + 0.356i)8-s + (−0.207 + 0.158i)10-s − 1.84i·11-s + (0.707 + 0.707i)13-s − 0.800·16-s + (−0.923 − 0.121i)20-s + (−0.341 + 0.341i)22-s + (−0.965 − 0.258i)25-s − 0.261i·26-s + (0.504 + 0.504i)32-s + (0.307 + 0.400i)40-s + 0.765i·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $-0.632 + 0.774i$
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: \((2,\ 1755,\ (\ :0),\ -0.632 + 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9718025985\)
\(L(\frac12)\) \(\approx\) \(0.9718025985\)
\(L(1)\) 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 \( 1 + (-0.130 + 0.991i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (0.184 + 0.184i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + 1.84iT - T^{2} \)
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 \( 1 - 0.765iT - T^{2} \)
43 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
47 \( 1 + (1.40 + 1.40i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 1.21T + T^{2} \)
61 \( 1 + 0.517T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.98iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (-0.860 + 0.860i)T - iT^{2} \)
89 \( 1 - 1.58T + T^{2} \)
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

−9.095921730325723694469409247064, −8.681055318410323883658885778668, −7.930567016111853985767065926558, −6.47792266064267767956434620490, −5.98973225918640066469695723900, −5.23650259102814773934960219667, −4.33066557107678818914820540167, −3.20382122243833734202848452541, −1.78952141939093959957837989447, −0.805676040075781848123001415600, 2.00311909714292648520444589831, 2.95867574011745242729205268736, 3.83318808489951535976571352711, 4.73331040591609894697696649799, 6.00957946428215193587519107343, 6.76427071243502614071471565759, 7.52261291114119688467975445652, 7.888794912920960234137066227181, 9.068972546780197927594836317846, 9.658810097323763766949985562966

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