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Properties

Label 2-28e2-28.27-c3-0-35
Degree $2$
Conductor $784$
Sign $0.944 - 0.327i$
Analytic cond. $46.2574$
Root an. cond. $6.80128$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.06·3-s + 1.22i·5-s + 22.8·9-s + 4.57i·11-s − 41.0i·13-s + 8.64i·15-s + 119. i·17-s + 60.1·19-s + 8.35i·23-s + 123.·25-s − 29.0·27-s + 164.·29-s + 287.·31-s + 32.3i·33-s − 148.·37-s + ⋯
L(s)  = 1  + 1.35·3-s + 0.109i·5-s + 0.847·9-s + 0.125i·11-s − 0.876i·13-s + 0.148i·15-s + 1.70i·17-s + 0.726·19-s + 0.0757i·23-s + 0.988·25-s − 0.207·27-s + 1.05·29-s + 1.66·31-s + 0.170i·33-s − 0.660·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.944 - 0.327i$
Analytic conductor: \(46.2574\)
Root analytic conductor: \(6.80128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :3/2),\ 0.944 - 0.327i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.528981624\)
\(L(\frac12)\) \(\approx\) \(3.528981624\)
\(L(\frac{5}{2})\) 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 \( 1 - 7.06T + 27T^{2} \)
5 \( 1 - 1.22iT - 125T^{2} \)
11 \( 1 - 4.57iT - 1.33e3T^{2} \)
13 \( 1 + 41.0iT - 2.19e3T^{2} \)
17 \( 1 - 119. iT - 4.91e3T^{2} \)
19 \( 1 - 60.1T + 6.85e3T^{2} \)
23 \( 1 - 8.35iT - 1.21e4T^{2} \)
29 \( 1 - 164.T + 2.43e4T^{2} \)
31 \( 1 - 287.T + 2.97e4T^{2} \)
37 \( 1 + 148.T + 5.06e4T^{2} \)
41 \( 1 - 358. iT - 6.89e4T^{2} \)
43 \( 1 + 360. iT - 7.95e4T^{2} \)
47 \( 1 - 225.T + 1.03e5T^{2} \)
53 \( 1 - 684.T + 1.48e5T^{2} \)
59 \( 1 - 85.0T + 2.05e5T^{2} \)
61 \( 1 + 209. iT - 2.26e5T^{2} \)
67 \( 1 - 417. iT - 3.00e5T^{2} \)
71 \( 1 - 982. iT - 3.57e5T^{2} \)
73 \( 1 + 341. iT - 3.89e5T^{2} \)
79 \( 1 + 76.2iT - 4.93e5T^{2} \)
83 \( 1 + 523.T + 5.71e5T^{2} \)
89 \( 1 + 972. iT - 7.04e5T^{2} \)
97 \( 1 + 676. iT - 9.12e5T^{2} \)
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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

−10.08428624869447480734281446466, −8.779082740308858237971848222039, −8.412374358314258513664760966936, −7.61380366927917008997879438958, −6.60330636897385294105358979569, −5.49486321288949559391451685676, −4.23519966769488392785329692927, −3.26697315709471011436554128730, −2.49788035714214858410299019745, −1.15367858243237165787230583014, 0.949824214590563073319197796965, 2.42926168752512248159070437664, 3.08496325542176949473423456970, 4.27175507733154495878521667193, 5.20371693911439541437277001576, 6.65073741075357364022749681060, 7.36663867371674654655165036788, 8.300286760053253150667885083960, 9.016651248526341035961121900173, 9.552115302982111150255963771838

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