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Properties

Label 2-3456-9.7-c1-0-34
Degree $2$
Conductor $3456$
Sign $0.766 + 0.642i$
Analytic cond. $27.5962$
Root an. cond. $5.25321$
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 + 1.73i)5-s + (1 − 1.73i)7-s + (2.5 − 4.33i)11-s + (2 + 3.46i)13-s − 17-s − 5·19-s + (−2 − 3.46i)23-s + (0.500 − 0.866i)25-s + (3 − 5.19i)29-s + 3.99·35-s + 10·37-s + (−1.5 − 2.59i)41-s + (4.5 − 7.79i)43-s + (4 − 6.92i)47-s + (1.50 + 2.59i)49-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + (0.377 − 0.654i)7-s + (0.753 − 1.30i)11-s + (0.554 + 0.960i)13-s − 0.242·17-s − 1.14·19-s + (−0.417 − 0.722i)23-s + (0.100 − 0.173i)25-s + (0.557 − 0.964i)29-s + 0.676·35-s + 1.64·37-s + (−0.234 − 0.405i)41-s + (0.686 − 1.18i)43-s + (0.583 − 1.01i)47-s + (0.214 + 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(27.5962\)
Root analytic conductor: \(5.25321\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.168388835\)
\(L(\frac12)\) \(\approx\) \(2.168388835\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.5 + 7.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 13T + 73T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (6.5 - 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.622243640217625205116015442073, −7.79465442283706258175114160744, −6.78053139235370071351585700756, −6.35754482215220907583779690393, −5.79020256698348401661416937421, −4.30303908181843879181687790199, −4.06214931400043204472946297993, −2.85081364345630671498651876544, −1.95982299473493753730593481228, −0.71222161688236074267577428335, 1.20408332334990641601245617759, 1.95723836376538868426065789389, 3.03411874414231672232268729271, 4.29764163923728929948656611390, 4.76888448954989546938937373951, 5.71141203825086631217451424797, 6.26508234801877561710814016015, 7.22755973575714988429169852543, 8.104145717161857738514149461125, 8.657429198959582768699298407855

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