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Properties

Label 2-28e2-112.109-c1-0-52
Degree $2$
Conductor $784$
Sign $0.0497 + 0.998i$
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  + (−0.0190 − 1.41i)2-s + (2.27 − 0.608i)3-s + (−1.99 + 0.0539i)4-s + (1.49 + 0.400i)5-s + (−0.903 − 3.19i)6-s + (0.114 + 2.82i)8-s + (2.18 − 1.26i)9-s + (0.537 − 2.12i)10-s + (0.469 + 1.75i)11-s + (−4.50 + 1.33i)12-s + (1.49 + 1.49i)13-s + 3.63·15-s + (3.99 − 0.215i)16-s + (2.44 − 4.22i)17-s + (−1.82 − 3.07i)18-s + (1.88 − 7.02i)19-s + ⋯
L(s)  = 1  + (−0.0134 − 0.999i)2-s + (1.31 − 0.351i)3-s + (−0.999 + 0.0269i)4-s + (0.668 + 0.179i)5-s + (−0.368 − 1.30i)6-s + (0.0404 + 0.999i)8-s + (0.729 − 0.421i)9-s + (0.170 − 0.670i)10-s + (0.141 + 0.528i)11-s + (−1.30 + 0.386i)12-s + (0.415 + 0.415i)13-s + 0.939·15-s + (0.998 − 0.0539i)16-s + (0.592 − 1.02i)17-s + (−0.431 − 0.723i)18-s + (0.431 − 1.61i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0497 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0497 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.0497 + 0.998i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.0497 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76193 - 1.67629i\)
\(L(\frac12)\) \(\approx\) \(1.76193 - 1.67629i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0190 + 1.41i)T \)
7 \( 1 \)
good3 \( 1 + (-2.27 + 0.608i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-1.49 - 0.400i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.469 - 1.75i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.49 - 1.49i)T + 13iT^{2} \)
17 \( 1 + (-2.44 + 4.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.88 + 7.02i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-6.87 + 3.97i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.18 + 4.18i)T + 29iT^{2} \)
31 \( 1 + (4.73 - 8.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.67 - 0.449i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 10.6iT - 41T^{2} \)
43 \( 1 + (-2.27 + 2.27i)T - 43iT^{2} \)
47 \( 1 + (0.930 + 1.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.18 - 8.15i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.00 - 3.73i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.38 - 5.17i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.647 - 0.173i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.50iT - 71T^{2} \)
73 \( 1 + (8.75 + 5.05i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.90 - 6.77i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.73 + 8.73i)T + 83iT^{2} \)
89 \( 1 + (3.69 - 2.13i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−9.880801809575910934281830560229, −9.182298684014673642009386426796, −8.844633792354727713798894640421, −7.66371265052196278221605595599, −6.86305513239698689102854701595, −5.37374115443609948589633649247, −4.36243770815314061108028759481, −3.04108230812872485666166183842, −2.53834560487652643122294030115, −1.33423514586608513710315160840, 1.60505341508677733786845591128, 3.40672623411542959778421716732, 3.85947658723825862654960693202, 5.53909483776039833734374154496, 5.84898558408910375051007065105, 7.32348972737151679927750826047, 8.021691874597818871410607884222, 8.716551254660536722295291793201, 9.451339858657563260992931452070, 9.966205033721517002388880786643

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