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Properties

Label 2-322-7.4-c1-0-8
Degree $2$
Conductor $322$
Sign $-0.00170 + 0.999i$
Analytic cond. $2.57118$
Root an. cond. $1.60349$
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  + (−0.5 − 0.866i)2-s + (−1.28 + 2.23i)3-s + (−0.499 + 0.866i)4-s + (−0.663 − 1.14i)5-s + 2.57·6-s + (−1.46 − 2.20i)7-s + 0.999·8-s + (−1.81 − 3.15i)9-s + (−0.663 + 1.14i)10-s + (−1.04 + 1.81i)11-s + (−1.28 − 2.23i)12-s + 3.11·13-s + (−1.17 + 2.36i)14-s + 3.41·15-s + (−0.5 − 0.866i)16-s + (3.47 − 6.01i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.743 + 1.28i)3-s + (−0.249 + 0.433i)4-s + (−0.296 − 0.513i)5-s + 1.05·6-s + (−0.552 − 0.833i)7-s + 0.353·8-s + (−0.606 − 1.05i)9-s + (−0.209 + 0.363i)10-s + (−0.315 + 0.546i)11-s + (−0.371 − 0.644i)12-s + 0.862·13-s + (−0.315 + 0.632i)14-s + 0.882·15-s + (−0.125 − 0.216i)16-s + (0.842 − 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00170 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00170 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.00170 + 0.999i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.00170 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.386303 - 0.386963i\)
\(L(\frac12)\) \(\approx\) \(0.386303 - 0.386963i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (1.46 + 2.20i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (1.28 - 2.23i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.663 + 1.14i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.04 - 1.81i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.11T + 13T^{2} \)
17 \( 1 + (-3.47 + 6.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.88 + 6.72i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 1.15T + 29T^{2} \)
31 \( 1 + (-4.35 + 7.54i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.65 - 2.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.26T + 41T^{2} \)
43 \( 1 + 8.36T + 43T^{2} \)
47 \( 1 + (-1.74 - 3.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.12 - 3.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.59 - 7.95i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.01 + 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.63 + 6.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.41T + 71T^{2} \)
73 \( 1 + (-2.18 + 3.78i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.398 - 0.690i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 + (8.32 + 14.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.65T + 97T^{2} \)
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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

−11.22040153916537187092892718283, −10.44115013585907181165281289203, −9.751315994298724444603332212340, −9.007285304215757508335261545170, −7.72533496705778790704969292417, −6.43831655071267736612909000822, −4.88304133116254841435159823437, −4.36768867488444447139865524111, −3.10531674061861146930969354187, −0.50659055062977806152411770583, 1.56032278180990819691875708338, 3.45431407634302859287882390395, 5.56692136794743789089457760249, 6.17721503570417934309329097335, 6.81701264912073752432293596296, 8.091279991167851976640247672133, 8.531161458056081100725105427597, 10.16377033243232760768550466101, 10.91834808973993250254380920207, 12.04096293272405350477982316722

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