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Properties

Label 2-968-11.9-c1-0-15
Degree $2$
Conductor $968$
Sign $0.998 + 0.0475i$
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  + (1.65 − 1.20i)3-s + (−0.0506 − 0.155i)5-s + (2.83 + 2.05i)7-s + (0.367 − 1.13i)9-s + (−0.832 + 2.56i)13-s + (−0.271 − 0.197i)15-s + (2.34 + 7.20i)17-s + (3.73 − 2.71i)19-s + 7.16·21-s − 0.706·23-s + (4.02 − 2.92i)25-s + (1.14 + 3.52i)27-s + (−0.832 − 0.604i)29-s + (1.66 − 5.13i)31-s + (0.177 − 0.545i)35-s + ⋯
L(s)  = 1  + (0.956 − 0.694i)3-s + (−0.0226 − 0.0696i)5-s + (1.07 + 0.777i)7-s + (0.122 − 0.377i)9-s + (−0.230 + 0.710i)13-s + (−0.0700 − 0.0508i)15-s + (0.567 + 1.74i)17-s + (0.857 − 0.622i)19-s + 1.56·21-s − 0.147·23-s + (0.804 − 0.584i)25-s + (0.220 + 0.678i)27-s + (−0.154 − 0.112i)29-s + (0.299 − 0.922i)31-s + (0.0299 − 0.0921i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.998 + 0.0475i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ 0.998 + 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.46310 - 0.0585687i\)
\(L(\frac12)\) \(\approx\) \(2.46310 - 0.0585687i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-1.65 + 1.20i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.0506 + 0.155i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.83 - 2.05i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.832 - 2.56i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.34 - 7.20i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.73 + 2.71i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 0.706T + 23T^{2} \)
29 \( 1 + (0.832 + 0.604i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.66 + 5.13i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (9.38 + 6.81i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.227 - 0.165i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.31T + 43T^{2} \)
47 \( 1 + (4.71 - 3.42i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.05 + 9.38i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.5 - 1.81i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.14 + 6.59i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 1.91T + 67T^{2} \)
71 \( 1 + (-3.76 - 11.5i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (9.20 + 6.68i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.533 + 1.64i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.45 + 4.47i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 6.31T + 89T^{2} \)
97 \( 1 + (-2.16 + 6.66i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.858970028111191534498040241688, −8.804281144770360711783581133307, −8.407495904329871240111627736543, −7.69878517807599093887574433737, −6.81113849844138218587736394415, −5.67162252141052325557914150934, −4.75646290225691102157648802984, −3.51308319540895858338687516860, −2.28497860294183280899095824185, −1.58780209305428841894295135141, 1.22804742848347355352649270639, 2.89264035784606760821790912183, 3.53987707590331857280681031295, 4.76635124029240010367545745123, 5.29349056927902435379888230646, 6.93884458407114858938813642514, 7.65705567017240257219465824363, 8.363670062168469671018144043593, 9.202428102003444797065206686619, 10.04231892074522471319220165569

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