Properties

Label 2-1755-195.47-c0-0-0
Degree $2$
Conductor $1755$
Sign $0.966 - 0.256i$
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  − 1.41·2-s + 1.00·4-s + (−0.707 − 0.707i)5-s + 7-s + (1.00 + 1.00i)10-s + (0.707 + 0.707i)11-s + 13-s − 1.41·14-s − 0.999·16-s + (0.707 + 0.707i)17-s + (−1 + i)19-s + (−0.707 − 0.707i)20-s + (−1.00 − 1.00i)22-s + 1.00i·25-s − 1.41·26-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.00·4-s + (−0.707 − 0.707i)5-s + 7-s + (1.00 + 1.00i)10-s + (0.707 + 0.707i)11-s + 13-s − 1.41·14-s − 0.999·16-s + (0.707 + 0.707i)17-s + (−1 + i)19-s + (−0.707 − 0.707i)20-s + (−1.00 − 1.00i)22-s + 1.00i·25-s − 1.41·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(0.875859\)
Root analytic conductor: \(0.935873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1755} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1755,\ (\ :0),\ 0.966 - 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5784716532\)
\(L(\frac12)\) \(\approx\) \(0.5784716532\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 - T \)
good2 \( 1 + 1.41T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
17 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
19 \( 1 + (1 - i)T - iT^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (1 + i)T + iT^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
59 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
97 \( 1 + iT - T^{2} \)
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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

−9.405720790434907942929567760904, −8.556930551110096147404930128433, −8.161614390537045278900253634174, −7.63488712013468559705617189396, −6.62584457141504154067158440796, −5.55838622453264946999996681285, −4.38073870248856346466314826275, −3.85317508636110359006920424681, −1.87273442627874359239007591741, −1.22289624250756854025761807587, 0.886300269604360441991794777864, 2.13708036896424846212987182625, 3.45334542279553442180053000133, 4.36489223054557317960488353479, 5.54858590982354557428096285810, 6.79458235235264015242919902032, 7.18008772138738325752682319834, 8.173737243570453025354746091191, 8.612262909857310924564339110100, 9.182693989663691525896000805968

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