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Properties

Label 2-1380-92.91-c1-0-47
Degree $2$
Conductor $1380$
Sign $0.465 + 0.885i$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
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  + (−1.41 + 0.0379i)2-s − i·3-s + (1.99 − 0.107i)4-s − i·5-s + (0.0379 + 1.41i)6-s − 1.92·7-s + (−2.81 + 0.227i)8-s − 9-s + (0.0379 + 1.41i)10-s + 6.24·11-s + (−0.107 − 1.99i)12-s + 1.29·13-s + (2.71 − 0.0729i)14-s − 15-s + (3.97 − 0.428i)16-s − 0.0974i·17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0268i)2-s − 0.577i·3-s + (0.998 − 0.0536i)4-s − 0.447i·5-s + (0.0154 + 0.577i)6-s − 0.726·7-s + (−0.996 + 0.0804i)8-s − 0.333·9-s + (0.0119 + 0.447i)10-s + 1.88·11-s + (−0.0309 − 0.576i)12-s + 0.359·13-s + (0.726 − 0.0194i)14-s − 0.258·15-s + (0.994 − 0.107i)16-s − 0.0236i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.465 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.465 + 0.885i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.465 + 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.077918851\)
\(L(\frac12)\) \(\approx\) \(1.077918851\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0379i)T \)
3 \( 1 + iT \)
5 \( 1 + iT \)
23 \( 1 + (-1.99 - 4.35i)T \)
good7 \( 1 + 1.92T + 7T^{2} \)
11 \( 1 - 6.24T + 11T^{2} \)
13 \( 1 - 1.29T + 13T^{2} \)
17 \( 1 + 0.0974iT - 17T^{2} \)
19 \( 1 + 1.71T + 19T^{2} \)
29 \( 1 - 6.14T + 29T^{2} \)
31 \( 1 - 2.62iT - 31T^{2} \)
37 \( 1 + 9.78iT - 37T^{2} \)
41 \( 1 - 9.53T + 41T^{2} \)
43 \( 1 - 4.28T + 43T^{2} \)
47 \( 1 - 4.88iT - 47T^{2} \)
53 \( 1 + 2.10iT - 53T^{2} \)
59 \( 1 + 9.68iT - 59T^{2} \)
61 \( 1 + 6.28iT - 61T^{2} \)
67 \( 1 - 4.41T + 67T^{2} \)
71 \( 1 - 10.2iT - 71T^{2} \)
73 \( 1 + 6.93T + 73T^{2} \)
79 \( 1 + 7.77T + 79T^{2} \)
83 \( 1 - 2.75T + 83T^{2} \)
89 \( 1 + 7.14iT - 89T^{2} \)
97 \( 1 + 6.15iT - 97T^{2} \)
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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

−9.198122383163337199901020995688, −8.876638829828034205332044482628, −7.893648211200910644489335138410, −6.98491572389923111763189124697, −6.42209038860442658883879379978, −5.70672529617838543304071879884, −4.11592014122770432342453621235, −3.09340021988763940859897981495, −1.74776416243364622067965685477, −0.796215539978138279529451769094, 1.04679358890912781576905930395, 2.57392045466696586509497811120, 3.50187124219463710636084835678, 4.42184784783427744441100256709, 6.12122887859916763829655179115, 6.40354141616207965598759826575, 7.24445244914299346142703313458, 8.440224844828612580991593903020, 8.996039482409984781516937630209, 9.675318039836274227446411484671

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