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Properties

Label 2-28e2-7.2-c1-0-2
Degree $2$
Conductor $784$
Sign $-0.827 - 0.561i$
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  + (0.707 + 1.22i)3-s + (−1.41 + 2.44i)5-s + (0.500 − 0.866i)9-s + (3 + 5.19i)11-s − 5.65·13-s − 4·15-s + (−0.707 − 1.22i)17-s + (−2.12 + 3.67i)19-s + (2 − 3.46i)23-s + (−1.49 − 2.59i)25-s + 5.65·27-s − 6·29-s + (1.41 + 2.44i)31-s + (−4.24 + 7.34i)33-s + (−1 + 1.73i)37-s + ⋯
L(s)  = 1  + (0.408 + 0.707i)3-s + (−0.632 + 1.09i)5-s + (0.166 − 0.288i)9-s + (0.904 + 1.56i)11-s − 1.56·13-s − 1.03·15-s + (−0.171 − 0.297i)17-s + (−0.486 + 0.842i)19-s + (0.417 − 0.722i)23-s + (−0.299 − 0.519i)25-s + 1.08·27-s − 1.11·29-s + (0.254 + 0.439i)31-s + (−0.738 + 1.27i)33-s + (−0.164 + 0.284i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.827 - 0.561i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.363813 + 1.18402i\)
\(L(\frac12)\) \(\approx\) \(0.363813 + 1.18402i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.707 - 1.22i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.41 - 2.44i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.12 - 3.67i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-1.41 - 2.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.41T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (1.41 - 2.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.707 - 1.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.24 + 7.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-4.94 - 8.57i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.41T + 83T^{2} \)
89 \( 1 + (-2.12 + 3.67i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−10.35094690343739993936662366568, −9.870717961608266443960795712576, −9.204050872184633140067457873793, −8.004438640968316716691778672094, −6.96594605850064020659660019889, −6.75513517204758891880906428657, −4.95607364234421573364315980005, −4.17758234036967552063566905299, −3.31088761496055060832943371755, −2.13541542563644472842114558992, 0.58249807334463023756725824810, 1.96916513969427169632100411388, 3.36834764559573638413678631977, 4.52253925892192648000565464489, 5.37050376450037368543833309528, 6.65434438933845784107595272082, 7.47815467025652877712209993698, 8.317363466820994164036931024645, 8.840549014867357699648521613361, 9.723239193921118663626830371560

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