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Properties

Label 2-2548-13.5-c0-0-1
Degree $2$
Conductor $2548$
Sign $0.881 + 0.471i$
Analytic cond. $1.27161$
Root an. cond. $1.12766$
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  + 1.41·3-s + (0.707 − 0.707i)5-s + 1.00·9-s + (−0.707 + 0.707i)13-s + (1.00 − 1.00i)15-s − 1.41i·17-s + (0.707 − 0.707i)19-s − i·23-s − 29-s + (−0.707 + 0.707i)31-s + (1 + i)37-s + (−1.00 + 1.00i)39-s + i·43-s + (0.707 − 0.707i)45-s + (0.707 + 0.707i)47-s + ⋯
L(s)  = 1  + 1.41·3-s + (0.707 − 0.707i)5-s + 1.00·9-s + (−0.707 + 0.707i)13-s + (1.00 − 1.00i)15-s − 1.41i·17-s + (0.707 − 0.707i)19-s − i·23-s − 29-s + (−0.707 + 0.707i)31-s + (1 + i)37-s + (−1.00 + 1.00i)39-s + i·43-s + (0.707 − 0.707i)45-s + (0.707 + 0.707i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.881 + 0.471i$
Analytic conductor: \(1.27161\)
Root analytic conductor: \(1.12766\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (785, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :0),\ 0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.117169872\)
\(L(\frac12)\) \(\approx\) \(2.117169872\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
13 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 - 1.41T + T^{2} \)
5 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
11 \( 1 + iT^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + (1 - i)T - iT^{2} \)
71 \( 1 + (1 - i)T - iT^{2} \)
73 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
89 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
97 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
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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

−9.036449901136071275297841152563, −8.540076325582319403604566526151, −7.43292022374580725509501548629, −7.11203608918565685806731698020, −5.84463365836190152423035512698, −4.93849624922076652841012025673, −4.28816444566749275919479489651, −2.99989622638371912573610358766, −2.47852947228702082011571428318, −1.37124271307593974350886651711, 1.81678024406073891736278018308, 2.41530868358777968719648294845, 3.43487267265167992252872095127, 3.92624054978879261851738218880, 5.45227604154539325389210406615, 5.96000962558609064104567865223, 7.13260258721512848625763794384, 7.71436284554181993969117046501, 8.293709979583084834011616115125, 9.259595317374064530416293645582

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