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Properties

Label 2-28e2-7.2-c1-0-17
Degree $2$
Conductor $784$
Sign $-0.991 - 0.126i$
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  + (−1.41 − 2.44i)3-s + (1.41 − 2.44i)5-s + (−2.49 + 4.33i)9-s + (−2 − 3.46i)11-s + 2.82·13-s − 8·15-s + (−2.82 − 4.89i)17-s + (1.41 − 2.44i)19-s + (−1.49 − 2.59i)25-s + 5.65·27-s + 2·29-s + (2.82 + 4.89i)31-s + (−5.65 + 9.79i)33-s + (−5 + 8.66i)37-s + (−4.00 − 6.92i)39-s + ⋯
L(s)  = 1  + (−0.816 − 1.41i)3-s + (0.632 − 1.09i)5-s + (−0.833 + 1.44i)9-s + (−0.603 − 1.04i)11-s + 0.784·13-s − 2.06·15-s + (−0.685 − 1.18i)17-s + (0.324 − 0.561i)19-s + (−0.299 − 0.519i)25-s + 1.08·27-s + 0.371·29-s + (0.508 + 0.879i)31-s + (−0.984 + 1.70i)33-s + (−0.821 + 1.42i)37-s + (−0.640 − 1.10i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
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: \((2,\ 784,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0644185 + 1.01510i\)
\(L(\frac12)\) \(\approx\) \(0.0644185 + 1.01510i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + 2.44i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.41 + 2.44i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 + (2.82 + 4.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-2.82 - 4.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.65T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (2.82 - 4.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.41 - 2.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.07 + 12.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.65T + 97T^{2} \)
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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

−9.834068147368146015171310102766, −8.698039374095817700249505291910, −8.260954216280067232538894504814, −7.03986785392424284806571631429, −6.36219104339578296897381600332, −5.43933138638671895470052479897, −4.88760341803873286245698793861, −2.92962681727128631462039570471, −1.53454617662963567977193443293, −0.57869028615880613105835254746, 2.17477154628923005311713845606, 3.55972908980583426825362757868, 4.38046965216307404292258623915, 5.48865138604822597556330434656, 6.14589314338855186807137694312, 7.02078636387979086199643010283, 8.322961338348044462070714695804, 9.440470831772266843837051101946, 10.15177255797167025413408455635, 10.57911746355247026279336256714

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