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Properties

Label 2-28e2-112.109-c1-0-69
Degree $2$
Conductor $784$
Sign $-0.868 + 0.496i$
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 − i)2-s + (−0.5 + 0.133i)3-s − 2i·4-s + (0.866 + 0.232i)5-s + (−0.366 + 0.633i)6-s + (−2 − 2i)8-s + (−2.36 + 1.36i)9-s + (1.09 − 0.633i)10-s + (−0.767 − 2.86i)11-s + (0.267 + i)12-s + (−3.73 − 3.73i)13-s − 0.464·15-s − 4·16-s + (3.23 − 5.59i)17-s + (−1.00 + 3.73i)18-s + (−0.767 + 2.86i)19-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.288 + 0.0773i)3-s − i·4-s + (0.387 + 0.103i)5-s + (−0.149 + 0.258i)6-s + (−0.707 − 0.707i)8-s + (−0.788 + 0.455i)9-s + (0.347 − 0.200i)10-s + (−0.231 − 0.864i)11-s + (0.0773 + 0.288i)12-s + (−1.03 − 1.03i)13-s − 0.119·15-s − 16-s + (0.783 − 1.35i)17-s + (−0.235 + 0.879i)18-s + (−0.176 + 0.657i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.868 + 0.496i$
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: \((2,\ 784,\ (\ :1/2),\ -0.868 + 0.496i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.388621 - 1.46248i\)
\(L(\frac12)\) \(\approx\) \(0.388621 - 1.46248i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
7 \( 1 \)
good3 \( 1 + (0.5 - 0.133i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.866 - 0.232i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.767 + 2.86i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (3.73 + 3.73i)T + 13iT^{2} \)
17 \( 1 + (-3.23 + 5.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.767 - 2.86i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.86 + 2.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.267 + 0.267i)T + 29iT^{2} \)
31 \( 1 + (-1.86 + 3.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.13 + 0.303i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.92iT - 41T^{2} \)
43 \( 1 + (-6.46 + 6.46i)T - 43iT^{2} \)
47 \( 1 + (-2.13 - 3.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.06 + 3.96i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.03 - 11.3i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.86 - 6.96i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (4.96 - 1.33i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.535iT - 71T^{2} \)
73 \( 1 + (6.23 + 3.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.33 - 14.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.53 + 1.53i)T + 83iT^{2} \)
89 \( 1 + (-4.5 + 2.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.92T + 97T^{2} \)
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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

−10.20618089592118550923704453435, −9.417946272643206538011639692755, −8.285450699014582242254048300884, −7.25302140651435147809889745401, −5.83654941961445572290419551780, −5.56651303844292023445706830883, −4.57475902022925831963077548496, −3.09030419722235994894656151792, −2.50540134771306052259077697058, −0.59298399482092017882129336201, 2.09285056245063321429274669993, 3.37894304335812631357172279382, 4.56645386662333247504960503752, 5.35830575782952959973139761612, 6.22342487114465632536740976756, 7.02340333798904947879288566687, 7.83654912924555626287042505350, 8.942280735804585768655953311313, 9.569792823305982298526284037054, 10.77986410843567834394878651348

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