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Properties

Label 2-28e2-196.83-c1-0-6
Degree $2$
Conductor $784$
Sign $0.979 - 0.200i$
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  + (−0.0556 + 0.0697i)3-s + (−3.27 − 2.61i)5-s + (1.51 + 2.17i)7-s + (0.665 + 2.91i)9-s + (−1.29 − 0.296i)11-s + (1.55 + 0.354i)13-s + (0.364 − 0.0832i)15-s + (−0.356 − 0.739i)17-s + 5.85·19-s + (−0.235 − 0.0152i)21-s + (1.52 − 3.16i)23-s + (2.79 + 12.2i)25-s + (−0.481 − 0.232i)27-s + (0.642 − 0.309i)29-s + 9.57·31-s + ⋯
L(s)  = 1  + (−0.0321 + 0.0402i)3-s + (−1.46 − 1.16i)5-s + (0.571 + 0.820i)7-s + (0.221 + 0.972i)9-s + (−0.391 − 0.0893i)11-s + (0.431 + 0.0984i)13-s + (0.0941 − 0.0214i)15-s + (−0.0863 − 0.179i)17-s + 1.34·19-s + (−0.0514 − 0.00332i)21-s + (0.317 − 0.659i)23-s + (0.558 + 2.44i)25-s + (−0.0927 − 0.0446i)27-s + (0.119 − 0.0574i)29-s + 1.71·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.979 - 0.200i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.979 - 0.200i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24286 + 0.126074i\)
\(L(\frac12)\) \(\approx\) \(1.24286 + 0.126074i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-1.51 - 2.17i)T \)
good3 \( 1 + (0.0556 - 0.0697i)T + (-0.667 - 2.92i)T^{2} \)
5 \( 1 + (3.27 + 2.61i)T + (1.11 + 4.87i)T^{2} \)
11 \( 1 + (1.29 + 0.296i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (-1.55 - 0.354i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (0.356 + 0.739i)T + (-10.5 + 13.2i)T^{2} \)
19 \( 1 - 5.85T + 19T^{2} \)
23 \( 1 + (-1.52 + 3.16i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (-0.642 + 0.309i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 - 9.57T + 31T^{2} \)
37 \( 1 + (4.49 - 2.16i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (-8.02 - 6.39i)T + (9.12 + 39.9i)T^{2} \)
43 \( 1 + (-4.98 + 3.97i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-2.26 + 9.93i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (-5.11 - 2.46i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (-2.50 - 3.13i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (4.17 + 8.66i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 - 12.7iT - 67T^{2} \)
71 \( 1 + (3.72 - 7.73i)T + (-44.2 - 55.5i)T^{2} \)
73 \( 1 + (-4.61 + 1.05i)T + (65.7 - 31.6i)T^{2} \)
79 \( 1 + 3.67iT - 79T^{2} \)
83 \( 1 + (-2.48 - 10.8i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (13.2 - 3.03i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + 9.96iT - 97T^{2} \)
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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

−10.43845248867276104082657419900, −9.228413365473156283996156192281, −8.385522569942478510777810090947, −8.035852271240300846946130917205, −7.14238008545240829222851191781, −5.53250080208048181139335126210, −4.90348589525738111005084640807, −4.13952263923913081160324411916, −2.72069535100091066541781299037, −1.07216411798247218293459799045, 0.864817037722735177472302757967, 2.97720694564552839244076845529, 3.74203692040975255233033297667, 4.55453789874824219121094169944, 6.05140535920930185324864422821, 7.11072331283599040766982518733, 7.50274020508243192951929337450, 8.296421578720990625449211708333, 9.520153604462140269098168392081, 10.53129865146485423821470428352

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