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Properties

Label 2-28e2-16.5-c1-0-28
Degree $2$
Conductor $784$
Sign $0.00225 - 0.999i$
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  + (−0.411 + 1.35i)2-s + (−2.21 + 2.21i)3-s + (−1.66 − 1.11i)4-s + (0.393 + 0.393i)5-s + (−2.08 − 3.90i)6-s + (2.18 − 1.79i)8-s − 6.81i·9-s + (−0.693 + 0.370i)10-s + (2.22 + 2.22i)11-s + (6.14 − 1.21i)12-s + (3.16 − 3.16i)13-s − 1.74·15-s + (1.52 + 3.69i)16-s − 0.980·17-s + (9.22 + 2.80i)18-s + (5.26 − 5.26i)19-s + ⋯
L(s)  = 1  + (−0.290 + 0.956i)2-s + (−1.27 + 1.27i)3-s + (−0.830 − 0.556i)4-s + (0.175 + 0.175i)5-s + (−0.851 − 1.59i)6-s + (0.774 − 0.633i)8-s − 2.27i·9-s + (−0.219 + 0.117i)10-s + (0.671 + 0.671i)11-s + (1.77 − 0.350i)12-s + (0.877 − 0.877i)13-s − 0.449·15-s + (0.380 + 0.924i)16-s − 0.237·17-s + (2.17 + 0.661i)18-s + (1.20 − 1.20i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00225 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00225 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.00225 - 0.999i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.00225 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.592554 + 0.591221i\)
\(L(\frac12)\) \(\approx\) \(0.592554 + 0.591221i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.411 - 1.35i)T \)
7 \( 1 \)
good3 \( 1 + (2.21 - 2.21i)T - 3iT^{2} \)
5 \( 1 + (-0.393 - 0.393i)T + 5iT^{2} \)
11 \( 1 + (-2.22 - 2.22i)T + 11iT^{2} \)
13 \( 1 + (-3.16 + 3.16i)T - 13iT^{2} \)
17 \( 1 + 0.980T + 17T^{2} \)
19 \( 1 + (-5.26 + 5.26i)T - 19iT^{2} \)
23 \( 1 - 1.25iT - 23T^{2} \)
29 \( 1 + (-3.17 + 3.17i)T - 29iT^{2} \)
31 \( 1 - 4.43T + 31T^{2} \)
37 \( 1 + (-0.645 - 0.645i)T + 37iT^{2} \)
41 \( 1 - 1.21iT - 41T^{2} \)
43 \( 1 + (-0.966 - 0.966i)T + 43iT^{2} \)
47 \( 1 + 9.97T + 47T^{2} \)
53 \( 1 + (8.07 + 8.07i)T + 53iT^{2} \)
59 \( 1 + (1.81 + 1.81i)T + 59iT^{2} \)
61 \( 1 + (-2.58 + 2.58i)T - 61iT^{2} \)
67 \( 1 + (1.59 - 1.59i)T - 67iT^{2} \)
71 \( 1 + 0.934iT - 71T^{2} \)
73 \( 1 - 0.710iT - 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + (-6.77 + 6.77i)T - 83iT^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 - 3.03T + 97T^{2} \)
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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

−10.24852499054532738092511117060, −9.752204016763886054100349783109, −9.035891145133435557693521710092, −7.919254135285196054115257757504, −6.58996409937483613902681162953, −6.23697359549864638266284055327, −5.12212859624214619454772531932, −4.62232879022648290197060921783, −3.49751504217755230031649048254, −0.77101554628477259982192621989, 1.04697500816027365653043528663, 1.71095377929267256402306355029, 3.37445086166769063437639314866, 4.70866340259036764267666613744, 5.75059201637060938764579820538, 6.50217442097356066356334691325, 7.52209754830369709281919551999, 8.398859484542724329633040450127, 9.303086341475388861121884171568, 10.38445291956280366177574234919

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