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Properties

Label 2-1344-56.3-c1-0-13
Degree $2$
Conductor $1344$
Sign $0.750 - 0.660i$
Analytic cond. $10.7318$
Root an. cond. $3.27595$
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  + (−0.866 − 0.5i)3-s + (1.5 + 2.59i)5-s + (0.866 + 2.5i)7-s + (0.499 + 0.866i)9-s + (2.59 − 4.5i)11-s − 2·13-s − 3i·15-s + (6.92 − 4i)19-s + (0.500 − 2.59i)21-s + (5.19 − 3i)23-s + (−2 + 3.46i)25-s − 0.999i·27-s + 5.19i·29-s + (−4.33 + 7.5i)31-s + (−4.5 + 2.59i)33-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.670 + 1.16i)5-s + (0.327 + 0.944i)7-s + (0.166 + 0.288i)9-s + (0.783 − 1.35i)11-s − 0.554·13-s − 0.774i·15-s + (1.58 − 0.917i)19-s + (0.109 − 0.566i)21-s + (1.08 − 0.625i)23-s + (−0.400 + 0.692i)25-s − 0.192i·27-s + 0.964i·29-s + (−0.777 + 1.34i)31-s + (−0.783 + 0.452i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.750 - 0.660i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.750 - 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.765724963\)
\(L(\frac12)\) \(\approx\) \(1.765724963\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.866 - 2.5i)T \)
good5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.59 + 4.5i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.92 + 4i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.19iT - 29T^{2} \)
31 \( 1 + (4.33 - 7.5i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6 + 3.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 + 3.46T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 2.59i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.79 + 4.5i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.73 + 3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + (-12 - 6.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.33 + 2.5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−9.567875212156423915627198135760, −9.082618677398631460557424910301, −8.053399684642885006742837765501, −6.92822302179025306291649690109, −6.52972098814561628084356685023, −5.57237980297371339922157214016, −4.98629364900228994819996907925, −3.23878844749148992189770039777, −2.67010665810514257640451782535, −1.23742441255198566401799543493, 0.954253521048290563437669970918, 1.87288325000423193496618149023, 3.71470532984223540810900220095, 4.57897387449282440943153823373, 5.16964480273315225294284192440, 6.03260458992922437582051593995, 7.25057135616363583533759158945, 7.66220480474103060351770973077, 9.020640623023691060166762731059, 9.699490479781922038254609895710

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