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Properties

Label 2-322-161.45-c1-0-3
Degree $2$
Conductor $322$
Sign $-0.317 - 0.948i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.317 - 0.948i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ -0.317 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.971705 + 1.34961i\)
\(L(\frac12)\) \(\approx\) \(0.971705 + 1.34961i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.14 - 4.29i)T \)
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

−11.92631155073247949407071245390, −11.02277998249610225288042357919, −9.977799101048891421600104962978, −9.067204041082671792104211592322, −7.992381606334559893810204907470, −6.89480254178991839689657953313, −5.91977612733303960861875283698, −5.41478501747847091562783620658, −3.40315553335043166198899207820, −2.57772952109510380461745327915, 1.19115452752006809458059855517, 2.57287581791995300234680747862, 4.54108327621072963945543415647, 4.90628324189812626540195480029, 6.17062257772603054688908168059, 7.71072058528832797128597141268, 8.651623819610047867609527256093, 9.620163958593894608281948922013, 10.46263770462306428743572146633, 11.30921550035849751235213271055

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