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Properties

Label 2-28e2-112.37-c1-0-7
Degree $2$
Conductor $784$
Sign $-0.109 - 0.994i$
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 + 0.0372i)2-s + (−3.13 − 0.839i)3-s + (1.99 − 0.105i)4-s + (2.74 − 0.734i)5-s + (4.46 + 1.07i)6-s + (−2.81 + 0.223i)8-s + (6.51 + 3.76i)9-s + (−3.84 + 1.14i)10-s + (−1.15 + 4.32i)11-s + (−6.34 − 1.34i)12-s + (−0.558 + 0.558i)13-s − 9.20·15-s + (3.97 − 0.420i)16-s + (−1.09 − 1.89i)17-s + (−9.34 − 5.07i)18-s + (0.684 + 2.55i)19-s + ⋯
L(s)  = 1  + (−0.999 + 0.0263i)2-s + (−1.80 − 0.484i)3-s + (0.998 − 0.0526i)4-s + (1.22 − 0.328i)5-s + (1.82 + 0.436i)6-s + (−0.996 + 0.0788i)8-s + (2.17 + 1.25i)9-s + (−1.21 + 0.360i)10-s + (−0.349 + 1.30i)11-s + (−1.83 − 0.388i)12-s + (−0.154 + 0.154i)13-s − 2.37·15-s + (0.994 − 0.105i)16-s + (−0.266 − 0.460i)17-s + (−2.20 − 1.19i)18-s + (0.156 + 0.585i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.109 - 0.994i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.109 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.232775 + 0.259755i\)
\(L(\frac12)\) \(\approx\) \(0.232775 + 0.259755i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0372i)T \)
7 \( 1 \)
good3 \( 1 + (3.13 + 0.839i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-2.74 + 0.734i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.15 - 4.32i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.558 - 0.558i)T - 13iT^{2} \)
17 \( 1 + (1.09 + 1.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.684 - 2.55i)T + (-16.4 + 9.5i)T^{2} \)
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 \( 1 + (6.33 - 1.69i)T + (32.0 - 18.5i)T^{2} \)
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 \( 1 + (1.16 - 4.36i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.89 - 7.07i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.53 - 5.73i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (5.53 + 1.48i)T + (58.0 + 33.5i)T^{2} \)
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

−10.37356016451413653702663175425, −9.880894814796849513028775334859, −9.151149093498438562917288608694, −7.58985196347145594778947147352, −7.10886522947423195381307100461, −6.09012809869424778853148502014, −5.60478670847107879016904326239, −4.62449498670462184672062293149, −2.16691758295596978050517297613, −1.37689559450778243838542617597, 0.31165718451330324170144880572, 1.79885215667547952434160109855, 3.43604253378312007340001139714, 5.18157984163355041098020241089, 5.79116700923850331964528902943, 6.41867200482195713712173348134, 7.20608523567000954387728248617, 8.627567421082039399545562217372, 9.522672520440025877230671548032, 10.23725410312444245613326521035

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