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Properties

Label 2-1944-9.7-c1-0-20
Degree $2$
Conductor $1944$
Sign $0.939 + 0.342i$
Analytic cond. $15.5229$
Root an. cond. $3.93991$
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  + (0.173 + 0.300i)5-s + (0.673 − 1.16i)7-s + (0.0923 − 0.160i)11-s + (0.918 + 1.59i)13-s + 0.758·17-s + 1.94·19-s + (−2.28 − 3.96i)23-s + (2.43 − 4.22i)25-s + (0.233 − 0.405i)29-s + (2.35 + 4.08i)31-s + 0.467·35-s + 2.17·37-s + (4.67 + 8.10i)41-s + (4.69 − 8.13i)43-s + (−0.826 + 1.43i)47-s + ⋯
L(s)  = 1  + (0.0776 + 0.134i)5-s + (0.254 − 0.441i)7-s + (0.0278 − 0.0482i)11-s + (0.254 + 0.441i)13-s + 0.184·17-s + 0.445·19-s + (−0.476 − 0.825i)23-s + (0.487 − 0.845i)25-s + (0.0434 − 0.0752i)29-s + (0.423 + 0.733i)31-s + 0.0790·35-s + 0.356·37-s + (0.730 + 1.26i)41-s + (0.716 − 1.24i)43-s + (−0.120 + 0.208i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(15.5229\)
Root analytic conductor: \(3.93991\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.868807754\)
\(L(\frac12)\) \(\approx\) \(1.868807754\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.173 - 0.300i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.673 + 1.16i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.0923 + 0.160i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.918 - 1.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.758T + 17T^{2} \)
19 \( 1 - 1.94T + 19T^{2} \)
23 \( 1 + (2.28 + 3.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.233 + 0.405i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.35 - 4.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.17T + 37T^{2} \)
41 \( 1 + (-4.67 - 8.10i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.69 + 8.13i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.826 - 1.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.70T + 53T^{2} \)
59 \( 1 + (3.39 + 5.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.99 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.35 - 2.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 - 1.55T + 73T^{2} \)
79 \( 1 + (-6.27 + 10.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.28 + 9.15i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.32T + 89T^{2} \)
97 \( 1 + (-2.88 + 5.00i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.167037052659320904070481836684, −8.287307494739026904170462338968, −7.67122397652915544294916701905, −6.68431713862061269555752936752, −6.13729338735982218502303491975, −4.97369360524275960474346730748, −4.27949674229112828557390704637, −3.25908078925850132317097429225, −2.16846484951724104792763599020, −0.863029762999419421628976836771, 1.05788302518822553290197873251, 2.29720325165664490900739011859, 3.34098167306856885368379635096, 4.31170477638383614464548812251, 5.39344768633206933533676440937, 5.83092950203722524448197123150, 6.94297059861591843374971856672, 7.75374712392445245539508039354, 8.401619158909928682467194274035, 9.325465238055515876392538603514

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