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Properties

Label 2-28e2-112.53-c1-0-28
Degree $2$
Conductor $784$
Sign $0.122 - 0.992i$
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  + (−0.366 + 1.36i)2-s + (0.366 + 1.36i)3-s + (−1.73 − i)4-s + (0.366 − 1.36i)5-s − 2·6-s + (2 − 1.99i)8-s + (0.866 − 0.5i)9-s + (1.73 + i)10-s + (−1.36 + 0.366i)11-s + (0.732 − 2.73i)12-s + (1 − i)13-s + 2·15-s + (1.99 + 3.46i)16-s + (−1 + 1.73i)17-s + (0.366 + 1.36i)18-s + (4.09 + 1.09i)19-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (0.211 + 0.788i)3-s + (−0.866 − 0.5i)4-s + (0.163 − 0.610i)5-s − 0.816·6-s + (0.707 − 0.707i)8-s + (0.288 − 0.166i)9-s + (0.547 + 0.316i)10-s + (−0.411 + 0.110i)11-s + (0.211 − 0.788i)12-s + (0.277 − 0.277i)13-s + 0.516·15-s + (0.499 + 0.866i)16-s + (−0.242 + 0.420i)17-s + (0.0862 + 0.321i)18-s + (0.940 + 0.251i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.122 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.122 - 0.992i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.122 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08941 + 0.963420i\)
\(L(\frac12)\) \(\approx\) \(1.08941 + 0.963420i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
7 \( 1 \)
good3 \( 1 + (-0.366 - 1.36i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.366 + 1.36i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.36 - 0.366i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1 + i)T - 13iT^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.09 - 1.09i)T + (16.4 + 9.5i)T^{2} \)
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 \( 1 + (-1.09 + 4.09i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-5 - 5i)T + 43iT^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.83 + 1.83i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (4.09 - 1.09i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (12.2 + 3.29i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.83 + 6.83i)T + (-58.0 + 33.5i)T^{2} \)
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

−10.36018445717742483868492435431, −9.207208618906316105909360312100, −9.069243883754808237280334692727, −7.966329844439294894814638382756, −7.11386553298184052910643055818, −6.05211541274121437411088437142, −5.06237604054954969541819905666, −4.47329661958484746521338452563, −3.27753167638859460829500718492, −1.14414461070119456571443506797, 1.07587071498164173990890850287, 2.34333829899732633025400223447, 3.14594347745426050238038872269, 4.45738847966052316452135854193, 5.56925844996236010240781353349, 7.01672793883560665313982491579, 7.44691804331731301786404736189, 8.521639888884699986549164313065, 9.318229337073321615772612224252, 10.23981446819947406141108035374

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