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Properties

Label 2-322-161.68-c1-0-4
Degree $2$
Conductor $322$
Sign $0.429 - 0.902i$
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 + (0.0922 − 0.0532i)3-s + (−0.499 − 0.866i)4-s + (−1.78 + 3.09i)5-s − 0.106i·6-s + (−0.672 + 2.55i)7-s − 0.999·8-s + (−1.49 + 2.58i)9-s + (1.78 + 3.09i)10-s + (2.00 − 1.15i)11-s + (−0.0922 − 0.0532i)12-s + 1.89i·13-s + (1.88 + 1.86i)14-s + 0.380i·15-s + (−0.5 + 0.866i)16-s + (−0.602 − 1.04i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.0532 − 0.0307i)3-s + (−0.249 − 0.433i)4-s + (−0.799 + 1.38i)5-s − 0.0434i·6-s + (−0.254 + 0.967i)7-s − 0.353·8-s + (−0.498 + 0.862i)9-s + (0.565 + 0.978i)10-s + (0.605 − 0.349i)11-s + (−0.0266 − 0.0153i)12-s + 0.524i·13-s + (0.502 + 0.497i)14-s + 0.0983i·15-s + (−0.125 + 0.216i)16-s + (−0.146 − 0.253i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.429 - 0.902i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.429 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.945091 + 0.596763i\)
\(L(\frac12)\) \(\approx\) \(0.945091 + 0.596763i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.672 - 2.55i)T \)
23 \( 1 + (-2.64 + 3.99i)T \)
good3 \( 1 + (-0.0922 + 0.0532i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.78 - 3.09i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.00 + 1.15i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.89iT - 13T^{2} \)
17 \( 1 + (0.602 + 1.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.42 - 2.46i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 3.48T + 29T^{2} \)
31 \( 1 + (-2.40 + 1.38i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.48 + 2.58i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.1iT - 41T^{2} \)
43 \( 1 - 2.46iT - 43T^{2} \)
47 \( 1 + (-10.1 - 5.84i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.58 + 4.37i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.57 - 2.64i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.78 + 4.82i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.2 - 5.90i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.295T + 71T^{2} \)
73 \( 1 + (-3.87 + 2.23i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.15 - 4.70i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.05T + 83T^{2} \)
89 \( 1 + (-2.84 + 4.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−11.64472302968540756292496720654, −11.07681318701775871698477476912, −10.25644385517588902862836208555, −9.012892543588695832254798906665, −8.086627818649794637099304629122, −6.80463220588036244382856906068, −5.95041823582645155714026759989, −4.50665387091124121998391273102, −3.21414019968349163069021178907, −2.39238038569114773970692172017, 0.72771580023753207505863273070, 3.54344055105106640040137422952, 4.31198499441677519561216404102, 5.37384951189203499140325282620, 6.68647478614612808046950373463, 7.56855888813519706475188130695, 8.670400773961911915476343057198, 9.170987834182185536560207307917, 10.53657358445780227368040432506, 11.89364138605169651723246958733

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