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Properties

Label 2-28e2-16.5-c1-0-75
Degree $2$
Conductor $784$
Sign $-0.540 + 0.841i$
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  + (1.32 − 0.482i)2-s + (1.83 − 1.83i)3-s + (1.53 − 1.28i)4-s + (−2.13 − 2.13i)5-s + (1.55 − 3.33i)6-s + (1.42 − 2.44i)8-s − 3.75i·9-s + (−3.86 − 1.80i)10-s + (2.28 + 2.28i)11-s + (0.462 − 5.17i)12-s + (−2.52 + 2.52i)13-s − 7.83·15-s + (0.708 − 3.93i)16-s + 0.402·17-s + (−1.81 − 4.99i)18-s + (1.01 − 1.01i)19-s + ⋯
L(s)  = 1  + (0.939 − 0.341i)2-s + (1.06 − 1.06i)3-s + (0.767 − 0.641i)4-s + (−0.953 − 0.953i)5-s + (0.635 − 1.35i)6-s + (0.502 − 0.864i)8-s − 1.25i·9-s + (−1.22 − 0.570i)10-s + (0.690 + 0.690i)11-s + (0.133 − 1.49i)12-s + (−0.699 + 0.699i)13-s − 2.02·15-s + (0.177 − 0.984i)16-s + 0.0975·17-s + (−0.427 − 1.17i)18-s + (0.233 − 0.233i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.540 + 0.841i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.540 + 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59096 - 2.91217i\)
\(L(\frac12)\) \(\approx\) \(1.59096 - 2.91217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 + 0.482i)T \)
7 \( 1 \)
good3 \( 1 + (-1.83 + 1.83i)T - 3iT^{2} \)
5 \( 1 + (2.13 + 2.13i)T + 5iT^{2} \)
11 \( 1 + (-2.28 - 2.28i)T + 11iT^{2} \)
13 \( 1 + (2.52 - 2.52i)T - 13iT^{2} \)
17 \( 1 - 0.402T + 17T^{2} \)
19 \( 1 + (-1.01 + 1.01i)T - 19iT^{2} \)
23 \( 1 - 9.11iT - 23T^{2} \)
29 \( 1 + (-1.47 + 1.47i)T - 29iT^{2} \)
31 \( 1 - 4.25T + 31T^{2} \)
37 \( 1 + (1.42 + 1.42i)T + 37iT^{2} \)
41 \( 1 + 8.96iT - 41T^{2} \)
43 \( 1 + (0.997 + 0.997i)T + 43iT^{2} \)
47 \( 1 + 4.19T + 47T^{2} \)
53 \( 1 + (-1.33 - 1.33i)T + 53iT^{2} \)
59 \( 1 + (-1.73 - 1.73i)T + 59iT^{2} \)
61 \( 1 + (-1.87 + 1.87i)T - 61iT^{2} \)
67 \( 1 + (-8.52 + 8.52i)T - 67iT^{2} \)
71 \( 1 - 0.451iT - 71T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + (0.742 - 0.742i)T - 83iT^{2} \)
89 \( 1 - 12.8iT - 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−9.812002881535862721476051008974, −9.126134625687456060174018044411, −8.107014618180614624813859832476, −7.31811810680231637388258851935, −6.79090688996764983473111111596, −5.32139645113922608019443791373, −4.32107322725514960766560724478, −3.51456511712761894587876356498, −2.22445075207103795057863416559, −1.23306448981155822643854612686, 2.74716087880358944818843019461, 3.20169320748789079154055151625, 4.08834883906250503780467405923, 4.84086201562813928620793684404, 6.24448649349367700220000077318, 7.10338029116576209741209561714, 8.131833491578275057175882939694, 8.525149932022557596885005094717, 9.904137695547862169013681848665, 10.59206841652017572890590462278

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