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Properties

Label 2-1264-316.103-c1-0-36
Degree $2$
Conductor $1264$
Sign $-0.904 + 0.425i$
Analytic cond. $10.0930$
Root an. cond. $3.17696$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.30 − 2.26i)3-s + (0.0495 + 0.0858i)5-s + (−0.841 − 1.45i)7-s + (−1.91 − 3.32i)9-s + (−2.75 + 1.58i)11-s + (2.72 − 4.71i)13-s + 0.259·15-s − 3.09i·17-s + (−1.45 + 0.838i)19-s − 4.40·21-s + (−5.97 + 3.45i)23-s + (2.49 − 4.32i)25-s − 2.18·27-s + (−1.50 + 0.871i)29-s + (−1.55 + 0.899i)31-s + ⋯
L(s)  = 1  + (0.754 − 1.30i)3-s + (0.0221 + 0.0383i)5-s + (−0.318 − 0.550i)7-s + (−0.639 − 1.10i)9-s + (−0.830 + 0.479i)11-s + (0.755 − 1.30i)13-s + 0.0669·15-s − 0.749i·17-s + (−0.333 + 0.192i)19-s − 0.960·21-s + (−1.24 + 0.719i)23-s + (0.499 − 0.864i)25-s − 0.421·27-s + (−0.280 + 0.161i)29-s + (−0.279 + 0.161i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1264\)    =    \(2^{4} \cdot 79\)
Sign: $-0.904 + 0.425i$
Analytic conductor: \(10.0930\)
Root analytic conductor: \(3.17696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1264} (735, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1264,\ (\ :1/2),\ -0.904 + 0.425i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.594976813\)
\(L(\frac12)\) \(\approx\) \(1.594976813\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
79 \( 1 + (-6.15 - 6.40i)T \)
good3 \( 1 + (-1.30 + 2.26i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.0495 - 0.0858i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.841 + 1.45i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.75 - 1.58i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.72 + 4.71i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.09iT - 17T^{2} \)
19 \( 1 + (1.45 - 0.838i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.97 - 3.45i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.50 - 0.871i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.55 - 0.899i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.60 + 2.08i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.90iT - 41T^{2} \)
43 \( 1 + (-1.98 + 3.44i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.0833 + 0.144i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.426 - 0.246i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.28 - 7.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 8.99iT - 61T^{2} \)
67 \( 1 + 15.2iT - 67T^{2} \)
71 \( 1 - 3.07T + 71T^{2} \)
73 \( 1 + (-0.753 - 1.30i)T + (-36.5 + 63.2i)T^{2} \)
83 \( 1 + (-5.11 + 2.95i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.59T + 89T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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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

−9.143852871557377337404326180319, −8.260782029554398238898111836713, −7.69338950655955467432908175986, −7.11686923178371894501455801718, −6.18509658507351246874644534257, −5.26018093016248053905169236125, −3.82657251375931996309083535054, −2.88948343139979640057604130638, −1.94259166995694030658559108800, −0.58112370106047476429696827404, 2.02927292710755731118591105996, 3.12194824458847991704389033688, 3.93642681574157898033221065004, 4.72114063097936323618207677960, 5.78465800483910805853337040515, 6.58690147398430407752980028520, 7.984030311246792815069261533108, 8.610834800712229383566730708538, 9.205410294991622113934723295124, 9.886446330897646200398984351505

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