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Properties

Label 2-28e2-112.53-c1-0-18
Degree $2$
Conductor $784$
Sign $0.569 - 0.821i$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
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  + (−0.953 + 1.04i)2-s + (−0.554 − 2.07i)3-s + (−0.182 − 1.99i)4-s + (−0.265 + 0.992i)5-s + (2.69 + 1.39i)6-s + (2.25 + 1.70i)8-s + (−1.38 + 0.797i)9-s + (−0.783 − 1.22i)10-s + (1.27 − 0.340i)11-s + (−4.02 + 1.48i)12-s + (−3.98 + 3.98i)13-s + 2.20·15-s + (−3.93 + 0.728i)16-s + (−2.81 + 4.87i)17-s + (0.483 − 2.20i)18-s + (4.69 + 1.25i)19-s + ⋯
L(s)  = 1  + (−0.673 + 0.738i)2-s + (−0.320 − 1.19i)3-s + (−0.0914 − 0.995i)4-s + (−0.118 + 0.443i)5-s + (1.09 + 0.569i)6-s + (0.797 + 0.603i)8-s + (−0.460 + 0.265i)9-s + (−0.247 − 0.386i)10-s + (0.383 − 0.102i)11-s + (−1.16 + 0.428i)12-s + (−1.10 + 1.10i)13-s + 0.568·15-s + (−0.983 + 0.182i)16-s + (−0.682 + 1.18i)17-s + (0.114 − 0.519i)18-s + (1.07 + 0.288i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.569 - 0.821i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.569 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.709336 + 0.371297i\)
\(L(\frac12)\) \(\approx\) \(0.709336 + 0.371297i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.953 - 1.04i)T \)
7 \( 1 \)
good3 \( 1 + (0.554 + 2.07i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.265 - 0.992i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.27 + 0.340i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (3.98 - 3.98i)T - 13iT^{2} \)
17 \( 1 + (2.81 - 4.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.69 - 1.25i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.495 - 0.286i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.81 + 5.81i)T - 29iT^{2} \)
31 \( 1 + (1.27 - 2.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.01 - 7.52i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 4.12iT - 41T^{2} \)
43 \( 1 + (-0.783 - 0.783i)T + 43iT^{2} \)
47 \( 1 + (-4.77 - 8.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.72 + 1.26i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-13.8 + 3.71i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (8.68 + 2.32i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.20 - 8.24i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.57iT - 71T^{2} \)
73 \( 1 + (-10.9 - 6.32i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.29 - 7.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.65 + 2.65i)T - 83iT^{2} \)
89 \( 1 + (2.44 - 1.41i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.98T + 97T^{2} \)
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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

−10.25824503703142162920186143126, −9.491807370127626374960319935556, −8.496118409907963722636779608340, −7.67417940944796199328364415787, −6.81883170714461117631178201784, −6.58199235758567918162137177313, −5.46913471592522863006471282566, −4.23113082259199430727187218313, −2.32178659431991165664547399160, −1.19388500049869986833754872688, 0.61200552671765548310255922408, 2.55531003822280772602103642002, 3.60224457141134782071997310182, 4.74739865608224465242722955110, 5.18375390429337338894598783938, 6.96718559082626223235741274309, 7.74711615929363747544888241909, 8.972757529817936365971836609888, 9.344241237372748189986042437706, 10.25867946240051589473797468931

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