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Properties

Label 2-28e2-112.53-c1-0-20
Degree $2$
Conductor $784$
Sign $-0.355 - 0.934i$
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  + (0.674 − 1.24i)2-s + (0.839 + 3.13i)3-s + (−1.08 − 1.67i)4-s + (−0.734 + 2.74i)5-s + (4.46 + 1.07i)6-s + (−2.81 + 0.223i)8-s + (−6.51 + 3.76i)9-s + (2.91 + 2.76i)10-s + (4.32 − 1.15i)11-s + (4.33 − 4.82i)12-s + (−0.558 + 0.558i)13-s − 9.20·15-s + (−1.62 + 3.65i)16-s + (−1.09 + 1.89i)17-s + (0.279 + 10.6i)18-s + (−2.55 − 0.684i)19-s + ⋯
L(s)  = 1  + (0.477 − 0.878i)2-s + (0.484 + 1.80i)3-s + (−0.544 − 0.838i)4-s + (−0.328 + 1.22i)5-s + (1.82 + 0.436i)6-s + (−0.996 + 0.0788i)8-s + (−2.17 + 1.25i)9-s + (0.920 + 0.873i)10-s + (1.30 − 0.349i)11-s + (1.25 − 1.39i)12-s + (−0.154 + 0.154i)13-s − 2.37·15-s + (−0.406 + 0.913i)16-s + (−0.266 + 0.460i)17-s + (0.0659 + 2.50i)18-s + (−0.585 − 0.156i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.355 - 0.934i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.355 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.950948 + 1.37835i\)
\(L(\frac12)\) \(\approx\) \(0.950948 + 1.37835i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.674 + 1.24i)T \)
7 \( 1 \)
good3 \( 1 + (-0.839 - 3.13i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.734 - 2.74i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.32 + 1.15i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.558 - 0.558i)T - 13iT^{2} \)
17 \( 1 + (1.09 - 1.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.55 + 0.684i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.647 + 0.374i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.07 - 3.07i)T - 29iT^{2} \)
31 \( 1 + (4.43 - 7.67i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.69 + 6.33i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.267iT - 41T^{2} \)
43 \( 1 + (-4.53 - 4.53i)T + 43iT^{2} \)
47 \( 1 + (3.74 + 6.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.36 + 1.16i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-7.07 + 1.89i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.73 + 1.53i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.48 - 5.53i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.37iT - 71T^{2} \)
73 \( 1 + (-9.08 - 5.24i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.10 - 8.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.474 + 0.474i)T - 83iT^{2} \)
89 \( 1 + (-12.5 + 7.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−10.73234111292411829231587901657, −9.912547010146344898521137663236, −9.108811251044585461267981413482, −8.568054410740040316964728415221, −6.91257669429217888637873927611, −5.85258116457167743584873935211, −4.76708537947916213898210508323, −3.75334773474228253261616974331, −3.49408977383746146054476161324, −2.31382935356466436066931051908, 0.67848558536993760153150752181, 2.10991993325614215964744731458, 3.63509362842507311420521554088, 4.70329530108946800444098722805, 5.93665327932448275366123973335, 6.57789755871814843512620347501, 7.53324834052392327593226928700, 7.993268446958877203723331199846, 8.983053223675594546873135891722, 9.256300701062048997988840500530

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