Properties

Label 2-322-23.4-c1-0-3
Degree $2$
Conductor $322$
Sign $0.815 - 0.578i$
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  + (−0.415 − 0.909i)2-s + (−0.589 + 0.173i)3-s + (−0.654 + 0.755i)4-s + (3.36 + 2.16i)5-s + (0.402 + 0.464i)6-s + (−0.142 − 0.989i)7-s + (0.959 + 0.281i)8-s + (−2.20 + 1.41i)9-s + (0.569 − 3.95i)10-s + (−2.44 + 5.35i)11-s + (0.255 − 0.558i)12-s + (−0.0479 + 0.333i)13-s + (−0.841 + 0.540i)14-s + (−2.35 − 0.691i)15-s + (−0.142 − 0.989i)16-s + (1.96 + 2.27i)17-s + ⋯
L(s)  = 1  + (−0.293 − 0.643i)2-s + (−0.340 + 0.0999i)3-s + (−0.327 + 0.377i)4-s + (1.50 + 0.966i)5-s + (0.164 + 0.189i)6-s + (−0.0537 − 0.374i)7-s + (0.339 + 0.0996i)8-s + (−0.735 + 0.472i)9-s + (0.179 − 1.25i)10-s + (−0.736 + 1.61i)11-s + (0.0736 − 0.161i)12-s + (−0.0132 + 0.0924i)13-s + (−0.224 + 0.144i)14-s + (−0.608 − 0.178i)15-s + (−0.0355 − 0.247i)16-s + (0.477 + 0.551i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.815 - 0.578i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.815 - 0.578i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06004 + 0.337478i\)
\(L(\frac12)\) \(\approx\) \(1.06004 + 0.337478i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 + 0.909i)T \)
7 \( 1 + (0.142 + 0.989i)T \)
23 \( 1 + (-3.84 + 2.86i)T \)
good3 \( 1 + (0.589 - 0.173i)T + (2.52 - 1.62i)T^{2} \)
5 \( 1 + (-3.36 - 2.16i)T + (2.07 + 4.54i)T^{2} \)
11 \( 1 + (2.44 - 5.35i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.0479 - 0.333i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-1.96 - 2.27i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-0.686 + 0.792i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (0.265 + 0.306i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (2.86 + 0.842i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-8.42 + 5.41i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-9.04 - 5.81i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (2.61 - 0.768i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 7.34T + 47T^{2} \)
53 \( 1 + (1.00 + 7.00i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-1.65 + 11.5i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (5.30 + 1.55i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-3.28 - 7.18i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (0.274 + 0.601i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (10.9 - 12.6i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-1.36 + 9.50i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (-4.62 + 2.97i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-6.07 + 1.78i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (2.44 + 1.57i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.34525432674816690462305376782, −10.68801070007354668348042570702, −9.993019528174763828024811775582, −9.403861803715435756711623964835, −7.900776350561975082530828412360, −6.87394839375628731502003918772, −5.76622552059708948203972116661, −4.71415933081512403364821386828, −2.88539862708241686746835170300, −1.98465639349360925864775579088, 0.959261644713516759794066414471, 2.90846200600161875694447966798, 5.12006183862043784727325399506, 5.69988961872761990416695383814, 6.25662418369106368435678438995, 7.87596848390307758206936832790, 8.923918148403552681374361427223, 9.291454514813420797346291418936, 10.45308987544120259347556444206, 11.47306808101525792212984681073

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