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Properties

Label 2-28e2-112.53-c1-0-3
Degree $2$
Conductor $784$
Sign $-0.401 - 0.915i$
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  + (−1.40 − 0.180i)2-s + (−0.152 − 0.569i)3-s + (1.93 + 0.505i)4-s + (0.414 − 1.54i)5-s + (0.111 + 0.826i)6-s + (−2.62 − 1.05i)8-s + (2.29 − 1.32i)9-s + (−0.859 + 2.09i)10-s + (−5.26 + 1.41i)11-s + (−0.00734 − 1.17i)12-s + (−4.66 + 4.66i)13-s − 0.943·15-s + (3.48 + 1.95i)16-s + (−2.66 + 4.61i)17-s + (−3.46 + 1.44i)18-s + (−3.49 − 0.936i)19-s + ⋯
L(s)  = 1  + (−0.991 − 0.127i)2-s + (−0.0880 − 0.328i)3-s + (0.967 + 0.252i)4-s + (0.185 − 0.691i)5-s + (0.0454 + 0.337i)6-s + (−0.927 − 0.374i)8-s + (0.765 − 0.442i)9-s + (−0.271 + 0.662i)10-s + (−1.58 + 0.425i)11-s + (−0.00212 − 0.340i)12-s + (−1.29 + 1.29i)13-s − 0.243·15-s + (0.872 + 0.489i)16-s + (−0.646 + 1.12i)17-s + (−0.815 + 0.340i)18-s + (−0.802 − 0.214i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.401 - 0.915i$
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: \((2,\ 784,\ (\ :1/2),\ -0.401 - 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.134754 + 0.206279i\)
\(L(\frac12)\) \(\approx\) \(0.134754 + 0.206279i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.180i)T \)
7 \( 1 \)
good3 \( 1 + (0.152 + 0.569i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.414 + 1.54i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (5.26 - 1.41i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (4.66 - 4.66i)T - 13iT^{2} \)
17 \( 1 + (2.66 - 4.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.49 + 0.936i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.25 + 1.30i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.22 - 1.22i)T - 29iT^{2} \)
31 \( 1 + (0.416 - 0.722i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.62 - 6.05i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.263iT - 41T^{2} \)
43 \( 1 + (-1.25 - 1.25i)T + 43iT^{2} \)
47 \( 1 + (5.37 + 9.31i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0650 - 0.0174i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (4.92 - 1.32i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-6.09 - 1.63i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-3.48 - 12.9i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.05iT - 71T^{2} \)
73 \( 1 + (-4.74 - 2.74i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.60 + 4.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.84 - 5.84i)T - 83iT^{2} \)
89 \( 1 + (5.47 - 3.16i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 18.8T + 97T^{2} \)
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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

−10.32587663269291195560576600223, −9.744384757500104760644042755130, −8.863120880791455449697134566112, −8.143379263754152924358892652748, −7.08963084531242370938375778555, −6.65922424606135817336651298553, −5.24009250659677296357923331537, −4.24585745588608582004888498298, −2.50201050834760247652944811043, −1.61134620631231821362513658924, 0.15921416130006297338847370227, 2.32265742745114219576720381707, 2.97516387356002401596953743433, 4.86113117757130900186540743581, 5.60701589755980877125698095239, 6.86338243156731255168248522151, 7.55740028631554111330773857964, 8.148610565113049583061275467335, 9.423856288315264244951707634451, 10.05648642453823043003291259384

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