Properties

Label 2-28e2-196.111-c1-0-18
Degree $2$
Conductor $784$
Sign $-0.439 + 0.898i$
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  + (−1.36 − 1.71i)3-s + (1.64 − 1.31i)5-s + (−1.20 + 2.35i)7-s + (−0.402 + 1.76i)9-s + (1.05 − 0.240i)11-s + (2.11 − 0.481i)13-s + (−4.50 − 1.02i)15-s + (3.50 − 7.27i)17-s + 4.92·19-s + (5.68 − 1.15i)21-s + (−3.41 − 7.10i)23-s + (−0.123 + 0.539i)25-s + (−2.35 + 1.13i)27-s + (−4.15 − 2.00i)29-s − 3.18·31-s + ⋯
L(s)  = 1  + (−0.789 − 0.990i)3-s + (0.737 − 0.587i)5-s + (−0.455 + 0.890i)7-s + (−0.134 + 0.588i)9-s + (0.317 − 0.0723i)11-s + (0.585 − 0.133i)13-s + (−1.16 − 0.265i)15-s + (0.849 − 1.76i)17-s + 1.12·19-s + (1.24 − 0.251i)21-s + (−0.713 − 1.48i)23-s + (−0.0246 + 0.107i)25-s + (−0.452 + 0.217i)27-s + (−0.771 − 0.371i)29-s − 0.572·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.439 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.439 + 0.898i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.439 + 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.631639 - 1.01231i\)
\(L(\frac12)\) \(\approx\) \(0.631639 - 1.01231i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1.20 - 2.35i)T \)
good3 \( 1 + (1.36 + 1.71i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (-1.64 + 1.31i)T + (1.11 - 4.87i)T^{2} \)
11 \( 1 + (-1.05 + 0.240i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (-2.11 + 0.481i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (-3.50 + 7.27i)T + (-10.5 - 13.2i)T^{2} \)
19 \( 1 - 4.92T + 19T^{2} \)
23 \( 1 + (3.41 + 7.10i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (4.15 + 2.00i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + 3.18T + 31T^{2} \)
37 \( 1 + (0.399 + 0.192i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (6.95 - 5.54i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (-3.49 - 2.78i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (1.14 + 5.00i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-0.514 + 0.247i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (-6.38 + 8.00i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (3.00 - 6.23i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + 11.8iT - 67T^{2} \)
71 \( 1 + (3.82 + 7.94i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-15.8 - 3.60i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + 3.47iT - 79T^{2} \)
83 \( 1 + (1.33 - 5.85i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (3.30 + 0.755i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + 10.5iT - 97T^{2} \)
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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

−9.718913293595513194251949454437, −9.379825663403567412874969196280, −8.286622335124671276369117468603, −7.24549649497615201967366910747, −6.37870722997272822113103202069, −5.66565156647856227556037474233, −5.07024199504329050887257788850, −3.27952481677266418585213904789, −1.94313710903683348441676284273, −0.70094796583011966365663121655, 1.57643508675876968048368220740, 3.52787799310591328551480957291, 3.98395350417562405399507390857, 5.49578269034636276683715156397, 5.90334775431713060155250678103, 6.94543662146629930765388632827, 7.911767387540061813035700968647, 9.293780942376827846941408200966, 9.987172474178486592425478117087, 10.42566452810879089410506344906

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