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Properties

Label 2-1568-7.4-c1-0-13
Degree $2$
Conductor $1568$
Sign $0.386 - 0.922i$
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  + (1 + 1.73i)5-s + (1.5 + 2.59i)9-s + 6·13-s + (−1 + 1.73i)17-s + (0.500 − 0.866i)25-s − 10·29-s + (1 + 1.73i)37-s + 10·41-s + (−3 + 5.19i)45-s + (−7 + 12.1i)53-s + (5 + 8.66i)61-s + (6 + 10.3i)65-s + (3 − 5.19i)73-s + (−4.5 + 7.79i)81-s − 3.99·85-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + (0.5 + 0.866i)9-s + 1.66·13-s + (−0.242 + 0.420i)17-s + (0.100 − 0.173i)25-s − 1.85·29-s + (0.164 + 0.284i)37-s + 1.56·41-s + (−0.447 + 0.774i)45-s + (−0.961 + 1.66i)53-s + (0.640 + 1.10i)61-s + (0.744 + 1.28i)65-s + (0.351 − 0.608i)73-s + (−0.5 + 0.866i)81-s − 0.433·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
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: \((2,\ 1568,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.982089203\)
\(L(\frac12)\) \(\approx\) \(1.982089203\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
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 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18T + 97T^{2} \)
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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

−9.596217089251235973144226146753, −8.816278662756817364832643584961, −7.912591218713821630998378719785, −7.20508973447611266448702088031, −6.22360093555634451850504371113, −5.73116554112964815885542667805, −4.47188546696729274756658672390, −3.61633248902851439760982642349, −2.47314716694803096831033800656, −1.45286753045269302624972575186, 0.854463060309764483263012365521, 1.86345545908322150849026141964, 3.42891139092492171791587163282, 4.12494590271738282204569281943, 5.24281242106899192112193828474, 6.00123654017432615011133482722, 6.75283449306363278982894449113, 7.74772030784588976463769790506, 8.690156043388332829496223141848, 9.270842932231006894609393718444

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