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Properties

Label 2-28e2-112.75-c1-0-60
Degree $2$
Conductor $784$
Sign $0.353 + 0.935i$
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  + (0.655 − 1.25i)2-s + (3.05 + 0.817i)3-s + (−1.14 − 1.64i)4-s + (−0.213 − 0.797i)5-s + (3.02 − 3.28i)6-s + (−2.80 + 0.353i)8-s + (6.05 + 3.49i)9-s + (−1.13 − 0.254i)10-s + (2.73 + 0.732i)11-s + (−2.13 − 5.94i)12-s + (−2.91 − 2.91i)13-s − 2.61i·15-s + (−1.39 + 3.74i)16-s + (2.30 − 1.33i)17-s + (8.34 − 5.29i)18-s + (−0.117 − 0.436i)19-s + ⋯
L(s)  = 1  + (0.463 − 0.886i)2-s + (1.76 + 0.472i)3-s + (−0.570 − 0.821i)4-s + (−0.0956 − 0.356i)5-s + (1.23 − 1.34i)6-s + (−0.992 + 0.125i)8-s + (2.01 + 1.16i)9-s + (−0.360 − 0.0806i)10-s + (0.823 + 0.220i)11-s + (−0.617 − 1.71i)12-s + (−0.807 − 0.807i)13-s − 0.673i·15-s + (−0.348 + 0.937i)16-s + (0.558 − 0.322i)17-s + (1.96 − 1.24i)18-s + (−0.0268 − 0.100i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.353 + 0.935i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.353 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.67546 - 1.84865i\)
\(L(\frac12)\) \(\approx\) \(2.67546 - 1.84865i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.655 + 1.25i)T \)
7 \( 1 \)
good3 \( 1 + (-3.05 - 0.817i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.213 + 0.797i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.73 - 0.732i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.91 + 2.91i)T + 13iT^{2} \)
17 \( 1 + (-2.30 + 1.33i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.117 + 0.436i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.63 + 2.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.04 - 2.04i)T - 29iT^{2} \)
31 \( 1 + (-1.26 - 2.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.33 - 0.625i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 11.9T + 41T^{2} \)
43 \( 1 + (3.27 - 3.27i)T - 43iT^{2} \)
47 \( 1 + (4.98 - 8.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.869 - 3.24i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.51 - 5.66i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.18 - 1.65i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.60 - 5.97i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.14T + 71T^{2} \)
73 \( 1 + (3.49 + 6.05i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.72 - 5.61i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.39 + 5.39i)T - 83iT^{2} \)
89 \( 1 + (0.528 - 0.915i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.2iT - 97T^{2} \)
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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

−10.03061245897545332616232580067, −9.343784085276534071298941311211, −8.712851658564660791617020411179, −7.907095614694205127249017836781, −6.74873245297663858571339078590, −5.09071771826858533021998121776, −4.45015878940255851117670135711, −3.36039419733889654201168862106, −2.74826104313597079713235838812, −1.45968889598379672270362460065, 1.89795553094800799942012623000, 3.24640534802275837395161340569, 3.76477725572442056965658152401, 4.99233520148897824732551387867, 6.53193081847099797735436500200, 7.04347347822399029119576374827, 7.82080491097770185107232711152, 8.594086561658091873414181309560, 9.265410171082717120980103462094, 9.959995295142695744237655250370

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