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Properties

Label 2-28e2-112.93-c1-0-34
Degree $2$
Conductor $784$
Sign $0.545 + 0.838i$
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  + (0.438 − 1.34i)2-s + (−0.0145 + 0.0543i)3-s + (−1.61 − 1.17i)4-s + (0.337 + 1.25i)5-s + (0.0667 + 0.0434i)6-s + (−2.29 + 1.65i)8-s + (2.59 + 1.49i)9-s + (1.83 + 0.0979i)10-s + (1.50 + 0.402i)11-s + (0.0876 − 0.0707i)12-s + (−1.59 − 1.59i)13-s − 0.0733·15-s + (1.22 + 3.80i)16-s + (−1.46 − 2.54i)17-s + (3.15 − 2.83i)18-s + (7.65 − 2.05i)19-s + ⋯
L(s)  = 1  + (0.309 − 0.950i)2-s + (−0.00841 + 0.0313i)3-s + (−0.807 − 0.589i)4-s + (0.150 + 0.562i)5-s + (0.0272 + 0.0177i)6-s + (−0.810 + 0.585i)8-s + (0.865 + 0.499i)9-s + (0.581 + 0.0309i)10-s + (0.453 + 0.121i)11-s + (0.0252 − 0.0204i)12-s + (−0.442 − 0.442i)13-s − 0.0189·15-s + (0.305 + 0.952i)16-s + (−0.356 − 0.617i)17-s + (0.742 − 0.667i)18-s + (1.75 − 0.470i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.545 + 0.838i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.545 + 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61148 - 0.874281i\)
\(L(\frac12)\) \(\approx\) \(1.61148 - 0.874281i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.438 + 1.34i)T \)
7 \( 1 \)
good3 \( 1 + (0.0145 - 0.0543i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.337 - 1.25i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.50 - 0.402i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.59 + 1.59i)T + 13iT^{2} \)
17 \( 1 + (1.46 + 2.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.65 + 2.05i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.91 - 2.26i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.06 - 2.06i)T + 29iT^{2} \)
31 \( 1 + (-3.14 - 5.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.40 + 5.24i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.34iT - 41T^{2} \)
43 \( 1 + (1.99 - 1.99i)T - 43iT^{2} \)
47 \( 1 + (0.979 - 1.69i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.2 + 3.00i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.96 - 0.793i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-9.72 + 2.60i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.566 + 2.11i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.26iT - 71T^{2} \)
73 \( 1 + (12.2 - 7.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.961 - 1.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.82 + 8.82i)T + 83iT^{2} \)
89 \( 1 + (-11.5 - 6.66i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.69T + 97T^{2} \)
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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

−10.21078401048544918823004269463, −9.618581047380312947095450213227, −8.770383480903875863571876211821, −7.44019681296591643507725260529, −6.75357803592446335148031220917, −5.31382436069016619001839353210, −4.74591198694075002234577570652, −3.44243306508105541908338437094, −2.58248582398048756304222342692, −1.18397489373478785472141365121, 1.17550518648142962141607635630, 3.21683647836463128833372023724, 4.33430645862149136854524917493, 5.02136547380205831556784999288, 6.15875557571520502129498772919, 6.87733543516493661970761154783, 7.72930025446013687197876754194, 8.669371297648497898155160431477, 9.463201744381086480202814024283, 10.00976843570680796615120551404

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