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Properties

Label 2-28e2-28.23-c2-0-23
Degree $2$
Conductor $784$
Sign $0.991 + 0.126i$
Analytic cond. $21.3624$
Root an. cond. $4.62195$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3 − 5.19i)5-s + (−4.5 + 7.79i)9-s + 10·13-s + (15 + 25.9i)17-s + (−5.5 − 9.52i)25-s + 42·29-s + (35 − 60.6i)37-s + 18·41-s + (27 + 46.7i)45-s + (−45 − 77.9i)53-s + (11 − 19.0i)61-s + (30 − 51.9i)65-s + (55 + 95.2i)73-s + (−40.5 − 70.1i)81-s + 180·85-s + ⋯
L(s)  = 1  + (0.600 − 1.03i)5-s + (−0.5 + 0.866i)9-s + 0.769·13-s + (0.882 + 1.52i)17-s + (−0.220 − 0.381i)25-s + 1.44·29-s + (0.945 − 1.63i)37-s + 0.439·41-s + (0.599 + 1.03i)45-s + (−0.849 − 1.47i)53-s + (0.180 − 0.312i)61-s + (0.461 − 0.799i)65-s + (0.753 + 1.30i)73-s + (−0.5 − 0.866i)81-s + 2.11·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(21.3624\)
Root analytic conductor: \(4.62195\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1),\ 0.991 + 0.126i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.146084442\)
\(L(\frac12)\) \(\approx\) \(2.146084442\)
\(L(2)\) 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 \( 1 \)
good3 \( 1 + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-3 + 5.19i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (60.5 - 104. i)T^{2} \)
13 \( 1 - 10T + 169T^{2} \)
17 \( 1 + (-15 - 25.9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (180.5 + 312. i)T^{2} \)
23 \( 1 + (264.5 + 458. i)T^{2} \)
29 \( 1 - 42T + 841T^{2} \)
31 \( 1 + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-35 + 60.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 18T + 1.68e3T^{2} \)
43 \( 1 - 1.84e3T^{2} \)
47 \( 1 + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (45 + 77.9i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-11 + 19.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + (-55 - 95.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 + (-39 + 67.5i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 130T + 9.40e3T^{2} \)
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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

−10.11701940161381503206478561663, −9.111011237622212222603713234665, −8.391881867742730668177142530101, −7.80095456283513911469104899658, −6.30963272153728957927775097266, −5.63220321205859869637177564896, −4.80195629271779627893171960406, −3.65280321706523944398747545527, −2.17879284127395676547427362987, −1.05874492856321622591574955477, 0.970450715513013931827555980606, 2.71783951901457985374976884574, 3.28259128474151795464031377461, 4.73591319102313764920776074097, 5.98287985220305806738995656057, 6.45722272628474700970444801345, 7.40930688754228980710432289127, 8.461162513157943756067335920892, 9.435670122925682345327210056715, 10.03414892617557391615728889633

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