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Properties

Label 2-1248-1248.1013-c0-0-1
Degree $2$
Conductor $1248$
Sign $0.707 + 0.707i$
Analytic cond. $0.622833$
Root an. cond. $0.789197$
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  + (−0.980 − 0.195i)2-s + (−0.923 + 0.382i)3-s + (0.923 + 0.382i)4-s + (0.750 − 1.81i)5-s + (0.980 − 0.195i)6-s + (−0.831 − 0.555i)8-s + (0.707 − 0.707i)9-s + (−1.08 + 1.63i)10-s + (1.53 + 0.636i)11-s − 12-s + (0.382 + 0.923i)13-s + 1.96i·15-s + (0.707 + 0.707i)16-s + (−0.831 + 0.555i)18-s + (1.38 − 1.38i)20-s + ⋯
L(s)  = 1  + (−0.980 − 0.195i)2-s + (−0.923 + 0.382i)3-s + (0.923 + 0.382i)4-s + (0.750 − 1.81i)5-s + (0.980 − 0.195i)6-s + (−0.831 − 0.555i)8-s + (0.707 − 0.707i)9-s + (−1.08 + 1.63i)10-s + (1.53 + 0.636i)11-s − 12-s + (0.382 + 0.923i)13-s + 1.96i·15-s + (0.707 + 0.707i)16-s + (−0.831 + 0.555i)18-s + (1.38 − 1.38i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (1013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1248,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6619657820\)
\(L(\frac12)\) \(\approx\) \(0.6619657820\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.980 + 0.195i)T \)
3 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + (-0.382 - 0.923i)T \)
good5 \( 1 + (-0.750 + 1.81i)T + (-0.707 - 0.707i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-1.53 - 0.636i)T + (0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.707 - 0.707i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (-0.707 + 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (-0.275 + 0.275i)T - iT^{2} \)
43 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + 1.11iT - T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.750 - 1.81i)T + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
67 \( 1 + (-0.707 + 0.707i)T^{2} \)
71 \( 1 + (1.17 + 1.17i)T + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 0.765iT - T^{2} \)
83 \( 1 + (0.149 + 0.360i)T + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (0.785 + 0.785i)T + iT^{2} \)
97 \( 1 - T^{2} \)
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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

−9.579185156596404804249147309207, −9.145667377607849041553660063314, −8.663122544390598254300396669735, −7.33355179516347241443413809423, −6.37952360064506717692088075341, −5.79808803163866614011135494224, −4.60453708353403496448334618322, −3.97788354355894980431212907004, −1.81903210563687473961761923482, −1.10929336102877170491212163656, 1.33079025287623517033398211612, 2.53705093331018259052645651506, 3.63320628244920589932171756492, 5.57483003495860364875866847576, 6.12966372555435395326903567496, 6.69368247373671736447099314717, 7.29878912002064848496012105746, 8.264748342737464644221680049515, 9.448386048550000023231719555897, 10.05877250057892325804666699262

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