Properties

Label 2-322-23.12-c1-0-7
Degree $2$
Conductor $322$
Sign $0.221 + 0.975i$
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  + (−0.841 + 0.540i)2-s + (0.0894 − 0.621i)3-s + (0.415 − 0.909i)4-s + (−2.40 + 0.705i)5-s + (0.260 + 0.571i)6-s + (−0.654 − 0.755i)7-s + (0.142 + 0.989i)8-s + (2.49 + 0.734i)9-s + (1.63 − 1.89i)10-s + (−2.09 − 1.34i)11-s + (−0.528 − 0.339i)12-s + (3.96 − 4.57i)13-s + (0.959 + 0.281i)14-s + (0.223 + 1.55i)15-s + (−0.654 − 0.755i)16-s + (−2.78 − 6.09i)17-s + ⋯
L(s)  = 1  + (−0.594 + 0.382i)2-s + (0.0516 − 0.359i)3-s + (0.207 − 0.454i)4-s + (−1.07 + 0.315i)5-s + (0.106 + 0.233i)6-s + (−0.247 − 0.285i)7-s + (0.0503 + 0.349i)8-s + (0.833 + 0.244i)9-s + (0.518 − 0.598i)10-s + (−0.632 − 0.406i)11-s + (−0.152 − 0.0980i)12-s + (1.09 − 1.26i)13-s + (0.256 + 0.0752i)14-s + (0.0577 + 0.401i)15-s + (−0.163 − 0.188i)16-s + (−0.675 − 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.221 + 0.975i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.221 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534131 - 0.426619i\)
\(L(\frac12)\) \(\approx\) \(0.534131 - 0.426619i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 - 0.540i)T \)
7 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (4.73 + 0.787i)T \)
good3 \( 1 + (-0.0894 + 0.621i)T + (-2.87 - 0.845i)T^{2} \)
5 \( 1 + (2.40 - 0.705i)T + (4.20 - 2.70i)T^{2} \)
11 \( 1 + (2.09 + 1.34i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-3.96 + 4.57i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (2.78 + 6.09i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-1.66 + 3.63i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (2.09 + 4.59i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-0.678 - 4.72i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (3.36 + 0.986i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (-11.2 + 3.29i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (0.533 - 3.70i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 - 4.86T + 47T^{2} \)
53 \( 1 + (-5.29 - 6.11i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (5.56 - 6.41i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (1.02 + 7.13i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-10.1 + 6.52i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (10.9 - 7.06i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-1.25 + 2.75i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (6.14 - 7.08i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (4.01 + 1.17i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (1.62 - 11.2i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (6.98 - 2.05i)T + (81.6 - 52.4i)T^{2} \)
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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

−11.14238189719498360855482934680, −10.63530082333616537942939745697, −9.500240774042290764006297490318, −8.299588238256506748249155574185, −7.60132372918266429541966825959, −6.96666432384413764185081493556, −5.67607868881576880189185524323, −4.24149460245986148084996816397, −2.83184648823220089887747881762, −0.61020235886100333387996028087, 1.76061874218304849549892542415, 3.78587071874868770397799884548, 4.22875561693179913686170664797, 6.07736506686397561964456223124, 7.28202937356970930726184746523, 8.226316763630459488317585311548, 9.003260433242256556472355629186, 9.989788652887685354554981815393, 10.84053276396532267987810494190, 11.75044414998618378000098061695

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