Properties

Label 2-722-361.138-c1-0-30
Degree $2$
Conductor $722$
Sign $-0.645 + 0.763i$
Analytic cond. $5.76519$
Root an. cond. $2.40108$
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  + (0.870 − 0.492i)2-s + (1.52 + 1.42i)3-s + (0.515 − 0.856i)4-s + (−4.34 − 0.159i)5-s + (2.02 + 0.493i)6-s + (−0.661 − 3.38i)7-s + (0.0275 − 0.999i)8-s + (0.0867 + 1.34i)9-s + (−3.86 + 1.99i)10-s + (−1.55 − 2.11i)11-s + (2.00 − 0.567i)12-s + (−5.72 − 2.44i)13-s + (−2.24 − 2.62i)14-s + (−6.38 − 6.43i)15-s + (−0.467 − 0.883i)16-s + (−0.929 + 0.0170i)17-s + ⋯
L(s)  = 1  + (0.615 − 0.347i)2-s + (0.878 + 0.823i)3-s + (0.257 − 0.428i)4-s + (−1.94 − 0.0714i)5-s + (0.827 + 0.201i)6-s + (−0.250 − 1.28i)7-s + (0.00974 − 0.353i)8-s + (0.0289 + 0.448i)9-s + (−1.22 + 0.631i)10-s + (−0.467 − 0.636i)11-s + (0.579 − 0.163i)12-s + (−1.58 − 0.679i)13-s + (−0.599 − 0.701i)14-s + (−1.64 − 1.66i)15-s + (−0.116 − 0.220i)16-s + (−0.225 + 0.00414i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.645 + 0.763i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ -0.645 + 0.763i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.470822 - 1.01515i\)
\(L(\frac12)\) \(\approx\) \(0.470822 - 1.01515i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.870 + 0.492i)T \)
19 \( 1 + (-1.69 - 4.01i)T \)
good3 \( 1 + (-1.52 - 1.42i)T + (0.192 + 2.99i)T^{2} \)
5 \( 1 + (4.34 + 0.159i)T + (4.98 + 0.367i)T^{2} \)
7 \( 1 + (0.661 + 3.38i)T + (-6.48 + 2.63i)T^{2} \)
11 \( 1 + (1.55 + 2.11i)T + (-3.28 + 10.4i)T^{2} \)
13 \( 1 + (5.72 + 2.44i)T + (8.97 + 9.40i)T^{2} \)
17 \( 1 + (0.929 - 0.0170i)T + (16.9 - 0.624i)T^{2} \)
23 \( 1 + (1.99 + 0.604i)T + (19.1 + 12.7i)T^{2} \)
29 \( 1 + (-4.03 - 3.25i)T + (6.08 + 28.3i)T^{2} \)
31 \( 1 + (0.280 + 0.748i)T + (-23.3 + 20.3i)T^{2} \)
37 \( 1 + (0.249 - 0.569i)T + (-25.0 - 27.2i)T^{2} \)
41 \( 1 + (-4.34 - 0.399i)T + (40.3 + 7.49i)T^{2} \)
43 \( 1 + (0.308 + 6.71i)T + (-42.8 + 3.94i)T^{2} \)
47 \( 1 + (8.17 - 8.56i)T + (-2.15 - 46.9i)T^{2} \)
53 \( 1 + (3.24 + 11.0i)T + (-44.6 + 28.5i)T^{2} \)
59 \( 1 + (0.717 + 1.55i)T + (-38.3 + 44.8i)T^{2} \)
61 \( 1 + (-2.70 + 0.553i)T + (56.0 - 23.9i)T^{2} \)
67 \( 1 + (11.4 - 1.70i)T + (64.1 - 19.4i)T^{2} \)
71 \( 1 + (0.972 + 0.199i)T + (65.2 + 27.9i)T^{2} \)
73 \( 1 + (-7.63 + 13.8i)T + (-38.7 - 61.8i)T^{2} \)
79 \( 1 + (-12.0 + 7.69i)T + (33.0 - 71.7i)T^{2} \)
83 \( 1 + (-0.124 + 0.899i)T + (-79.8 - 22.5i)T^{2} \)
89 \( 1 + (-0.156 - 0.282i)T + (-47.3 + 75.3i)T^{2} \)
97 \( 1 + (1.07 + 0.159i)T + (92.8 + 28.1i)T^{2} \)
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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

−10.35851715691916083165302135356, −9.386723203316693802812322778001, −8.086152425028539385641042612643, −7.77468039515231509564159937885, −6.76661373343673928732830852469, −5.00071578605614567319362712085, −4.28995556507376933226689450315, −3.52005629299000495344731591558, −2.99111297245818356245670563767, −0.40490870961264523577968869993, 2.40797062709392321404065173578, 2.96066986138945883660960206474, 4.35748223543894706588778789456, 5.07719153774821823173475807360, 6.70517145063931248468177783746, 7.32606489745480540805929592157, 7.922197202573159060211882405296, 8.624811817405995380316362682870, 9.556468656905967305221052850774, 11.11093982204969200864554507418

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