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Properties

Label 2-322-7.4-c1-0-5
Degree $2$
Conductor $322$
Sign $-0.999 - 0.00612i$
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 + (−1.33 + 2.31i)3-s + (−0.499 + 0.866i)4-s + (1.93 + 3.34i)5-s − 2.67·6-s + (−2.21 + 1.45i)7-s − 0.999·8-s + (−2.08 − 3.60i)9-s + (−1.93 + 3.34i)10-s + (3.04 − 5.27i)11-s + (−1.33 − 2.31i)12-s + 5.97·13-s + (−2.36 − 1.18i)14-s − 10.3·15-s + (−0.5 − 0.866i)16-s + (−0.430 + 0.745i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.772 + 1.33i)3-s + (−0.249 + 0.433i)4-s + (0.863 + 1.49i)5-s − 1.09·6-s + (−0.835 + 0.548i)7-s − 0.353·8-s + (−0.693 − 1.20i)9-s + (−0.610 + 1.05i)10-s + (0.918 − 1.59i)11-s + (−0.386 − 0.669i)12-s + 1.65·13-s + (−0.631 − 0.317i)14-s − 2.66·15-s + (−0.125 − 0.216i)16-s + (−0.104 + 0.180i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.999 - 0.00612i$
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: \((2,\ 322,\ (\ :1/2),\ -0.999 - 0.00612i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00399916 + 1.30590i\)
\(L(\frac12)\) \(\approx\) \(0.00399916 + 1.30590i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 1.45i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (1.33 - 2.31i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.93 - 3.34i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.04 + 5.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.97T + 13T^{2} \)
17 \( 1 + (0.430 - 0.745i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.556 + 0.963i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 0.670T + 29T^{2} \)
31 \( 1 + (-2.11 + 3.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.71 - 2.96i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.45T + 41T^{2} \)
43 \( 1 + 3.73T + 43T^{2} \)
47 \( 1 + (-2.00 - 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.01 + 1.75i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.580 + 1.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.10 - 7.11i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0278 + 0.0482i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.863T + 71T^{2} \)
73 \( 1 + (-8.46 + 14.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.24 + 2.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.05T + 83T^{2} \)
89 \( 1 + (-3.44 - 5.96i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−11.66497222746308778294699359059, −11.10008244926900800074776947480, −10.31547535338338095051487179083, −9.402378513876487299627271152996, −8.569160230000663720411376910583, −6.50691482044200990339727298448, −6.24400455997992139147082279239, −5.53501108982531726916802690907, −3.78139892080747904847609973027, −3.17477298290424920751192504921, 1.03281277489778284240708581485, 1.81499270096518832370041502758, 4.04044139002256827569200709457, 5.22769808902430597323286003764, 6.26967351168746700310198325581, 6.88153077857243239305854787025, 8.451976059460012927152683026051, 9.417199562462212222820871330615, 10.26796234739066395936295215063, 11.55320958437373903295784167177

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