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Properties

Label 2-1424-356.247-c0-0-0
Degree $2$
Conductor $1424$
Sign $0.999 - 0.0364i$
Analytic cond. $0.710668$
Root an. cond. $0.843011$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.983 + 0.449i)5-s + (−0.540 − 0.841i)9-s + (0.373 + 0.203i)13-s + (1.29 − 0.186i)17-s + (0.110 + 0.127i)25-s + (0.133 − 0.0498i)29-s + (0.847 + 0.847i)37-s + (−0.841 + 0.459i)41-s + (−0.153 − 1.07i)45-s + (−0.755 + 0.654i)49-s + (−0.0801 − 0.273i)53-s + (0.0498 + 0.697i)61-s + (0.275 + 0.367i)65-s + (−1.61 − 1.03i)73-s + (−0.415 + 0.909i)81-s + ⋯
L(s)  = 1  + (0.983 + 0.449i)5-s + (−0.540 − 0.841i)9-s + (0.373 + 0.203i)13-s + (1.29 − 0.186i)17-s + (0.110 + 0.127i)25-s + (0.133 − 0.0498i)29-s + (0.847 + 0.847i)37-s + (−0.841 + 0.459i)41-s + (−0.153 − 1.07i)45-s + (−0.755 + 0.654i)49-s + (−0.0801 − 0.273i)53-s + (0.0498 + 0.697i)61-s + (0.275 + 0.367i)65-s + (−1.61 − 1.03i)73-s + (−0.415 + 0.909i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1424\)    =    \(2^{4} \cdot 89\)
Sign: $0.999 - 0.0364i$
Analytic conductor: \(0.710668\)
Root analytic conductor: \(0.843011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1424} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1424,\ (\ :0),\ 0.999 - 0.0364i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.290368583\)
\(L(\frac12)\) \(\approx\) \(1.290368583\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 + (0.415 + 0.909i)T \)
good3 \( 1 + (0.540 + 0.841i)T^{2} \)
5 \( 1 + (-0.983 - 0.449i)T + (0.654 + 0.755i)T^{2} \)
7 \( 1 + (0.755 - 0.654i)T^{2} \)
11 \( 1 + (0.654 - 0.755i)T^{2} \)
13 \( 1 + (-0.373 - 0.203i)T + (0.540 + 0.841i)T^{2} \)
17 \( 1 + (-1.29 + 0.186i)T + (0.959 - 0.281i)T^{2} \)
19 \( 1 + (-0.909 - 0.415i)T^{2} \)
23 \( 1 + (0.909 + 0.415i)T^{2} \)
29 \( 1 + (-0.133 + 0.0498i)T + (0.755 - 0.654i)T^{2} \)
31 \( 1 + (0.909 - 0.415i)T^{2} \)
37 \( 1 + (-0.847 - 0.847i)T + iT^{2} \)
41 \( 1 + (0.841 - 0.459i)T + (0.540 - 0.841i)T^{2} \)
43 \( 1 + (-0.755 - 0.654i)T^{2} \)
47 \( 1 + (0.841 + 0.540i)T^{2} \)
53 \( 1 + (0.0801 + 0.273i)T + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + (0.540 - 0.841i)T^{2} \)
61 \( 1 + (-0.0498 - 0.697i)T + (-0.989 + 0.142i)T^{2} \)
67 \( 1 + (-0.841 + 0.540i)T^{2} \)
71 \( 1 + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \)
79 \( 1 + (0.415 + 0.909i)T^{2} \)
83 \( 1 + (-0.281 - 0.959i)T^{2} \)
97 \( 1 + (-0.234 + 0.512i)T + (-0.654 - 0.755i)T^{2} \)
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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

−9.840988470926233187216705535980, −9.050790496777422326613104001656, −8.218155430548583458170592325226, −7.23314068212966368905824578228, −6.20944224158813464122906460094, −5.92758700768146475136220141187, −4.79513697554608245545123476070, −3.50230536694018160169063839979, −2.74269128532226215787183155356, −1.38748112221236075736364466438, 1.43476290761087334257144798259, 2.51212365096870653122257741187, 3.64995977660657360422106515855, 4.97035783661167217917175681229, 5.56010279413532934415528283064, 6.22947272367777559763651350617, 7.44430522404783109280136061267, 8.188114672505802855721113190845, 8.935411991033273159401963840684, 9.789238826895944027711543579587

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