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Properties

Label 2-1568-7.4-c1-0-10
Degree $2$
Conductor $1568$
Sign $-0.0725 - 0.997i$
Analytic cond. $12.5205$
Root an. cond. $3.53843$
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  + (0.707 + 1.22i)5-s + (1.5 + 2.59i)9-s + 1.41·13-s + (−3.53 + 6.12i)17-s + (1.50 − 2.59i)25-s − 4·29-s + (6 + 10.3i)37-s − 12.7·41-s + (−2.12 + 3.67i)45-s + (7 − 12.1i)53-s + (7.77 + 13.4i)61-s + (1.00 + 1.73i)65-s + (−7.77 + 13.4i)73-s + (−4.5 + 7.79i)81-s − 10·85-s + ⋯
L(s)  = 1  + (0.316 + 0.547i)5-s + (0.5 + 0.866i)9-s + 0.392·13-s + (−0.857 + 1.48i)17-s + (0.300 − 0.519i)25-s − 0.742·29-s + (0.986 + 1.70i)37-s − 1.98·41-s + (−0.316 + 0.547i)45-s + (0.961 − 1.66i)53-s + (0.995 + 1.72i)61-s + (0.124 + 0.214i)65-s + (−0.910 + 1.57i)73-s + (−0.5 + 0.866i)81-s − 1.08·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $-0.0725 - 0.997i$
Analytic conductor: \(12.5205\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ -0.0725 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.600567785\)
\(L(\frac12)\) \(\approx\) \(1.600567785\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.707 - 1.22i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 + (3.53 - 6.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6 - 10.3i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 12.7T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7 + 12.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.77 - 13.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (7.77 - 13.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-2.12 - 3.67i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.07T + 97T^{2} \)
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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

−9.969377617830624252498401685068, −8.592926660910084583091290835142, −8.244819974876687575866201999226, −7.08471684164886081882463951201, −6.50691925969973186132511540599, −5.60882942126336861255293283614, −4.59668520298099739830050770347, −3.72909355207045080813124426810, −2.50177370249410515091273590689, −1.57724015093110778731764293019, 0.62888233593511019042510827028, 1.90690990286574435919044436764, 3.19919112937038644031866050549, 4.20236882720751536753054404170, 5.06487378249606925017542920537, 5.93928705257255865428522676976, 6.87269813678533957402600429406, 7.48163848983583726250348846493, 8.699644061668985788862039937953, 9.215609673023474112002449016108

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