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Properties

Label 2-936-8.5-c1-0-18
Degree $2$
Conductor $936$
Sign $0.998 - 0.0585i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
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  + (−1.21 + 0.730i)2-s + (0.931 − 1.76i)4-s + 2.32i·5-s − 4.85·7-s + (0.165 + 2.82i)8-s + (−1.70 − 2.81i)10-s − 3.45i·11-s − i·13-s + (5.87 − 3.54i)14-s + (−2.26 − 3.29i)16-s − 0.641·17-s − 5.60i·19-s + (4.11 + 2.16i)20-s + (2.52 + 4.18i)22-s + 9.37·23-s + ⋯
L(s)  = 1  + (−0.856 + 0.516i)2-s + (0.465 − 0.884i)4-s + 1.04i·5-s − 1.83·7-s + (0.0585 + 0.998i)8-s + (−0.538 − 0.891i)10-s − 1.04i·11-s − 0.277i·13-s + (1.56 − 0.947i)14-s + (−0.566 − 0.824i)16-s − 0.155·17-s − 1.28i·19-s + (0.921 + 0.484i)20-s + (0.538 + 0.892i)22-s + 1.95·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.998 - 0.0585i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.998 - 0.0585i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.726867 + 0.0212994i\)
\(L(\frac12)\) \(\approx\) \(0.726867 + 0.0212994i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.21 - 0.730i)T \)
3 \( 1 \)
13 \( 1 + iT \)
good5 \( 1 - 2.32iT - 5T^{2} \)
7 \( 1 + 4.85T + 7T^{2} \)
11 \( 1 + 3.45iT - 11T^{2} \)
17 \( 1 + 0.641T + 17T^{2} \)
19 \( 1 + 5.60iT - 19T^{2} \)
23 \( 1 - 9.37T + 23T^{2} \)
29 \( 1 - 5.01iT - 29T^{2} \)
31 \( 1 - 2.89T + 31T^{2} \)
37 \( 1 - 5.74iT - 37T^{2} \)
41 \( 1 + 2.69T + 41T^{2} \)
43 \( 1 + 0.307iT - 43T^{2} \)
47 \( 1 - 8.30T + 47T^{2} \)
53 \( 1 + 1.60iT - 53T^{2} \)
59 \( 1 - 0.256iT - 59T^{2} \)
61 \( 1 + 11.1iT - 61T^{2} \)
67 \( 1 + 9.58iT - 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 - 5.74T + 73T^{2} \)
79 \( 1 - 8.80T + 79T^{2} \)
83 \( 1 - 13.4iT - 83T^{2} \)
89 \( 1 - 3.75T + 89T^{2} \)
97 \( 1 + 1.98T + 97T^{2} \)
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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

−9.964741451964836778231350672407, −9.192184374838773218348694361028, −8.599524539885076708867726799212, −7.24449842210663096590427575613, −6.74138261797009755580641288280, −6.22468174978716017025795059104, −5.11454257621846366157744371204, −3.19756583196626406423079028306, −2.82934106085460239397987776805, −0.60560032765931564972640256959, 0.934574052977436391258575201705, 2.40964267310049440225105648979, 3.54147633217152189384467427381, 4.50004744179401842151936048743, 5.90508403052005263460655343313, 6.87067332669737084323560588612, 7.55937090743025538812060949211, 8.832800092520791350230069201625, 9.173338039067092775678574934155, 9.964927275115341262998259863461

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