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Properties

Label 2-322-23.18-c1-0-2
Degree $2$
Conductor $322$
Sign $0.912 - 0.409i$
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  + (−0.959 − 0.281i)2-s + (−0.358 + 0.414i)3-s + (0.841 + 0.540i)4-s + (0.138 − 0.961i)5-s + (0.461 − 0.296i)6-s + (0.415 + 0.909i)7-s + (−0.654 − 0.755i)8-s + (0.384 + 2.67i)9-s + (−0.403 + 0.883i)10-s + (1.38 − 0.407i)11-s + (−0.525 + 0.154i)12-s + (0.233 − 0.511i)13-s + (−0.142 − 0.989i)14-s + (0.348 + 0.402i)15-s + (0.415 + 0.909i)16-s + (1.24 − 0.797i)17-s + ⋯
L(s)  = 1  + (−0.678 − 0.199i)2-s + (−0.207 + 0.239i)3-s + (0.420 + 0.270i)4-s + (0.0617 − 0.429i)5-s + (0.188 − 0.120i)6-s + (0.157 + 0.343i)7-s + (−0.231 − 0.267i)8-s + (0.128 + 0.890i)9-s + (−0.127 + 0.279i)10-s + (0.418 − 0.122i)11-s + (−0.151 + 0.0445i)12-s + (0.0647 − 0.141i)13-s + (−0.0380 − 0.264i)14-s + (0.0899 + 0.103i)15-s + (0.103 + 0.227i)16-s + (0.300 − 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.912 - 0.409i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.912 - 0.409i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.941509 + 0.201822i\)
\(L(\frac12)\) \(\approx\) \(0.941509 + 0.201822i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (-1.86 - 4.41i)T \)
good3 \( 1 + (0.358 - 0.414i)T + (-0.426 - 2.96i)T^{2} \)
5 \( 1 + (-0.138 + 0.961i)T + (-4.79 - 1.40i)T^{2} \)
11 \( 1 + (-1.38 + 0.407i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-0.233 + 0.511i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-1.24 + 0.797i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-2.83 - 1.82i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (-1.90 + 1.22i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (-5.49 - 6.34i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (-0.701 - 4.88i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (-1.64 + 11.4i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (1.83 - 2.11i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 - 0.380T + 47T^{2} \)
53 \( 1 + (3.18 + 6.97i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (1.60 - 3.51i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (3.42 + 3.94i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (12.9 + 3.81i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (-1.99 - 0.587i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (6.51 + 4.18i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (5.95 - 13.0i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (1.17 + 8.16i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (2.24 - 2.58i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (-1.82 + 12.6i)T + (-93.0 - 27.3i)T^{2} \)
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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

−11.59882234824685374929879778818, −10.66893800998960337932711709850, −9.851545748601478173408272475655, −8.908438686127242015609125570630, −8.074926280620588196422340000889, −7.07209204692936019904454977494, −5.69170091863367234988777267741, −4.72910778572237318974305753769, −3.11607923356942848073199038578, −1.46264601007710613758970354154, 1.06124234716524486439495430184, 2.93072963668497846490686472240, 4.46545182720924711752276213050, 6.06723773506056779804010338714, 6.76559246725333178224633835259, 7.64997589793091689762451193267, 8.818193916047087207977051018790, 9.662132033916667552562783580180, 10.56300938660286055967427776714, 11.47581201147833095090045772372

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