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Properties

Label 2-76-19.11-c5-0-5
Degree $2$
Conductor $76$
Sign $0.657 + 0.753i$
Analytic cond. $12.1891$
Root an. cond. $3.49129$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.95 − 13.7i)3-s + (−15.4 + 26.7i)5-s + 132.·7-s + (−4.91 − 8.50i)9-s + 670.·11-s + (−411. − 712. i)13-s + (245. + 425. i)15-s + (731. − 1.26e3i)17-s + (−1.57e3 − 30.7i)19-s + (1.05e3 − 1.82e3i)21-s + (−1.14e3 − 1.98e3i)23-s + (1.08e3 + 1.87e3i)25-s + 3.70e3·27-s + (−1.38e3 − 2.39e3i)29-s + 1.05e4·31-s + ⋯
L(s)  = 1  + (0.510 − 0.883i)3-s + (−0.276 + 0.479i)5-s + 1.01·7-s + (−0.0202 − 0.0350i)9-s + 1.67·11-s + (−0.675 − 1.16i)13-s + (0.282 + 0.488i)15-s + (0.613 − 1.06i)17-s + (−0.999 − 0.0195i)19-s + (0.520 − 0.900i)21-s + (−0.451 − 0.782i)23-s + (0.346 + 0.600i)25-s + 0.978·27-s + (−0.304 − 0.528i)29-s + 1.97·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.657 + 0.753i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ 0.657 + 0.753i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.12673 - 0.967539i\)
\(L(\frac12)\) \(\approx\) \(2.12673 - 0.967539i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (1.57e3 + 30.7i)T \)
good3 \( 1 + (-7.95 + 13.7i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (15.4 - 26.7i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 - 132.T + 1.68e4T^{2} \)
11 \( 1 - 670.T + 1.61e5T^{2} \)
13 \( 1 + (411. + 712. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-731. + 1.26e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
23 \( 1 + (1.14e3 + 1.98e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (1.38e3 + 2.39e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 - 1.05e4T + 2.86e7T^{2} \)
37 \( 1 - 4.81e3T + 6.93e7T^{2} \)
41 \( 1 + (7.28e3 - 1.26e4i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (5.15e3 - 8.92e3i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (2.55e3 + 4.42e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (3.72e3 + 6.45e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.71e4 - 2.97e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.06e4 - 3.56e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.96e4 + 3.39e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-5.84e3 + 1.01e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (1.36e4 - 2.37e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (2.58e3 - 4.47e3i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 8.61e4T + 3.93e9T^{2} \)
89 \( 1 + (-2.53e4 - 4.39e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + (1.33e4 - 2.30e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−13.49531741085963398442381224093, −12.24091097786899106178315090546, −11.41263865200708979659858164586, −10.02052984223791862114106117237, −8.441094059700934993297211433623, −7.62444042124464534262002259355, −6.51706462973447376136534251805, −4.62152994422794291010014920409, −2.74542042463096423334790257440, −1.18331880370352388859413522393, 1.58000472970328633973919405592, 3.90411255985853822355629965714, 4.61633603051902984261165974884, 6.56559695781049593198741075685, 8.273779528262917140689524163364, 9.075310537441791134973105378285, 10.14809867276914493585616925994, 11.58299809012205577165422568674, 12.32876999276931416866704774762, 14.21034883595273077457027424887

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