Properties

Label 2-722-1.1-c1-0-5
Degree $2$
Conductor $722$
Sign $1$
Analytic cond. $5.76519$
Root an. cond. $2.40108$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.53·3-s + 4-s + 2·5-s + 1.53·6-s + 2.69·7-s − 8-s − 0.652·9-s − 2·10-s + 3.18·11-s − 1.53·12-s + 5.75·13-s − 2.69·14-s − 3.06·15-s + 16-s − 6.51·17-s + 0.652·18-s + 2·20-s − 4.12·21-s − 3.18·22-s − 0.694·23-s + 1.53·24-s − 25-s − 5.75·26-s + 5.59·27-s + 2.69·28-s + 2.82·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.884·3-s + 0.5·4-s + 0.894·5-s + 0.625·6-s + 1.01·7-s − 0.353·8-s − 0.217·9-s − 0.632·10-s + 0.960·11-s − 0.442·12-s + 1.59·13-s − 0.720·14-s − 0.791·15-s + 0.250·16-s − 1.58·17-s + 0.153·18-s + 0.447·20-s − 0.900·21-s − 0.679·22-s − 0.144·23-s + 0.312·24-s − 0.200·25-s − 1.12·26-s + 1.07·27-s + 0.509·28-s + 0.524·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.106992466\)
\(L(\frac12)\) \(\approx\) \(1.106992466\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
19 \( 1 \)
good3 \( 1 + 1.53T + 3T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
7 \( 1 - 2.69T + 7T^{2} \)
11 \( 1 - 3.18T + 11T^{2} \)
13 \( 1 - 5.75T + 13T^{2} \)
17 \( 1 + 6.51T + 17T^{2} \)
23 \( 1 + 0.694T + 23T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + 2.45T + 31T^{2} \)
37 \( 1 - 4.36T + 37T^{2} \)
41 \( 1 - 0.347T + 41T^{2} \)
43 \( 1 - 6.06T + 43T^{2} \)
47 \( 1 - 7.88T + 47T^{2} \)
53 \( 1 + 8.21T + 53T^{2} \)
59 \( 1 + 0.573T + 59T^{2} \)
61 \( 1 - 2.93T + 61T^{2} \)
67 \( 1 - 4.95T + 67T^{2} \)
71 \( 1 - 8.45T + 71T^{2} \)
73 \( 1 - 15.7T + 73T^{2} \)
79 \( 1 - 9.06T + 79T^{2} \)
83 \( 1 - 8.47T + 83T^{2} \)
89 \( 1 + 7.73T + 89T^{2} \)
97 \( 1 - 0.347T + 97T^{2} \)
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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

−10.70608456391033715892821252605, −9.418073384910046762245221473982, −8.816405147039691545437046672649, −8.010489372792661589567420402807, −6.55166341077350550769152703559, −6.22164820933960032027253201021, −5.22886524434935809283915471190, −4.02210022282092120725833698475, −2.20934183984285395708508559718, −1.10910445578557446570450185485, 1.10910445578557446570450185485, 2.20934183984285395708508559718, 4.02210022282092120725833698475, 5.22886524434935809283915471190, 6.22164820933960032027253201021, 6.55166341077350550769152703559, 8.010489372792661589567420402807, 8.816405147039691545437046672649, 9.418073384910046762245221473982, 10.70608456391033715892821252605

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