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Properties

Label 2-28e2-112.37-c1-0-71
Degree $2$
Conductor $784$
Sign $0.868 + 0.496i$
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  + (1.36 − 0.366i)2-s + (1.36 + 0.366i)3-s + (1.73 − i)4-s + (1.36 − 0.366i)5-s + 2·6-s + (1.99 − 2i)8-s + (−0.866 − 0.5i)9-s + (1.73 − i)10-s + (0.366 − 1.36i)11-s + (2.73 − 0.732i)12-s + (−1 + i)13-s + 2·15-s + (1.99 − 3.46i)16-s + (1 + 1.73i)17-s + (−1.36 − 0.366i)18-s + (1.09 + 4.09i)19-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.788 + 0.211i)3-s + (0.866 − 0.5i)4-s + (0.610 − 0.163i)5-s + 0.816·6-s + (0.707 − 0.707i)8-s + (−0.288 − 0.166i)9-s + (0.547 − 0.316i)10-s + (0.110 − 0.411i)11-s + (0.788 − 0.211i)12-s + (−0.277 + 0.277i)13-s + 0.516·15-s + (0.499 − 0.866i)16-s + (0.242 + 0.420i)17-s + (−0.321 − 0.0862i)18-s + (0.251 + 0.940i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.868 + 0.496i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.868 + 0.496i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.65607 - 0.971517i\)
\(L(\frac12)\) \(\approx\) \(3.65607 - 0.971517i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
7 \( 1 \)
good3 \( 1 + (-1.36 - 0.366i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-1.36 + 0.366i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.366 + 1.36i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (1 - i)T - 13iT^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.09 - 4.09i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 3i)T - 29iT^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.09 - 1.09i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-5 - 5i)T + 43iT^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.83 - 6.83i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.09 - 4.09i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.29 + 12.2i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-6.83 - 1.83i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 10iT - 71T^{2} \)
73 \( 1 + (3.46 - 2i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1 - i)T - 83iT^{2} \)
89 \( 1 + (3.46 + 2i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.11998327460161857023804498481, −9.617257946805794096672637987174, −8.506505521826283961670871692464, −7.70800209695716241270447798871, −6.32087354075610996175627755323, −5.86351615621741669134261876481, −4.65042219647258950486118344057, −3.65876892468198362399949278965, −2.77174750763484560347845687918, −1.63568432097551402479036568706, 2.03398092580905579983917475861, 2.75791260802413482393592779634, 3.86174722923435438824818566210, 5.07311198676537972605702712708, 5.84135120913581840798836824872, 6.89111219209469923537613511063, 7.66066447397198738362196793308, 8.443917634660731925796462734698, 9.528731008776832739257337163995, 10.33765986444270539211088462236

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