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Properties

Label 2-968-8.5-c1-0-95
Degree $2$
Conductor $968$
Sign $0.152 - 0.988i$
Analytic cond. $7.72951$
Root an. cond. $2.78020$
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  + (0.0720 − 1.41i)2-s − 1.81i·3-s + (−1.98 − 0.203i)4-s − 3.73i·5-s + (−2.55 − 0.130i)6-s − 1.19·7-s + (−0.430 + 2.79i)8-s − 0.282·9-s + (−5.27 − 0.269i)10-s + (−0.368 + 3.60i)12-s + 4.68i·13-s + (−0.0857 + 1.68i)14-s − 6.76·15-s + (3.91 + 0.809i)16-s − 5.57·17-s + (−0.0203 + 0.399i)18-s + ⋯
L(s)  = 1  + (0.0509 − 0.998i)2-s − 1.04i·3-s + (−0.994 − 0.101i)4-s − 1.67i·5-s + (−1.04 − 0.0532i)6-s − 0.449·7-s + (−0.152 + 0.988i)8-s − 0.0942·9-s + (−1.66 − 0.0850i)10-s + (−0.106 + 1.04i)12-s + 1.29i·13-s + (−0.0229 + 0.449i)14-s − 1.74·15-s + (0.979 + 0.202i)16-s − 1.35·17-s + (−0.00480 + 0.0941i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.152 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.152 - 0.988i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ 0.152 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.570032 + 0.488951i\)
\(L(\frac12)\) \(\approx\) \(0.570032 + 0.488951i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0720 + 1.41i)T \)
11 \( 1 \)
good3 \( 1 + 1.81iT - 3T^{2} \)
5 \( 1 + 3.73iT - 5T^{2} \)
7 \( 1 + 1.19T + 7T^{2} \)
13 \( 1 - 4.68iT - 13T^{2} \)
17 \( 1 + 5.57T + 17T^{2} \)
19 \( 1 + 3.83iT - 19T^{2} \)
23 \( 1 + 7.04T + 23T^{2} \)
29 \( 1 - 1.05iT - 29T^{2} \)
31 \( 1 - 3.47T + 31T^{2} \)
37 \( 1 - 1.78iT - 37T^{2} \)
41 \( 1 - 9.24T + 41T^{2} \)
43 \( 1 + 3.91iT - 43T^{2} \)
47 \( 1 - 8.75T + 47T^{2} \)
53 \( 1 + 3.93iT - 53T^{2} \)
59 \( 1 + 12.3iT - 59T^{2} \)
61 \( 1 - 3.62iT - 61T^{2} \)
67 \( 1 - 8.07iT - 67T^{2} \)
71 \( 1 + 9.67T + 71T^{2} \)
73 \( 1 + 6.96T + 73T^{2} \)
79 \( 1 + 7.71T + 79T^{2} \)
83 \( 1 + 7.46iT - 83T^{2} \)
89 \( 1 - 7.95T + 89T^{2} \)
97 \( 1 + 0.293T + 97T^{2} \)
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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

−9.227252497134881223881437379307, −8.796954956529240011235745438327, −7.947573179583454321858430033150, −6.79991554050829376929849115420, −5.83715918453634775757618213154, −4.56061257650815290149836341785, −4.18348885150423789420954573146, −2.34928240662024570657095609709, −1.55090700000777176749225270412, −0.34519818222123184268810123333, 2.75050939908989949908142487861, 3.71946438276774985736660630343, 4.40919901981374258259293312627, 5.80650184733739811452359819638, 6.26001091352713201761765721759, 7.28922540485119827870468507205, 7.936919516012424961584747689169, 9.077163006266630178625624326038, 9.999912033416691820660131498833, 10.31994798062813324517433932067

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