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Properties

Label 2-1380-92.91-c1-0-17
Degree $2$
Conductor $1380$
Sign $-0.993 + 0.110i$
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  + (0.847 + 1.13i)2-s − i·3-s + (−0.562 + 1.91i)4-s + i·5-s + (1.13 − 0.847i)6-s + 1.84·7-s + (−2.64 + 0.991i)8-s − 9-s + (−1.13 + 0.847i)10-s − 2.86·11-s + (1.91 + 0.562i)12-s − 4.13·13-s + (1.56 + 2.09i)14-s + 15-s + (−3.36 − 2.15i)16-s + 6.96i·17-s + ⋯
L(s)  = 1  + (0.599 + 0.800i)2-s − 0.577i·3-s + (−0.281 + 0.959i)4-s + 0.447i·5-s + (0.462 − 0.346i)6-s + 0.698·7-s + (−0.936 + 0.350i)8-s − 0.333·9-s + (−0.357 + 0.268i)10-s − 0.864·11-s + (0.554 + 0.162i)12-s − 1.14·13-s + (0.418 + 0.559i)14-s + 0.258·15-s + (−0.842 − 0.539i)16-s + 1.69i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.993 + 0.110i$
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: \((2,\ 1380,\ (\ :1/2),\ -0.993 + 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9883243916\)
\(L(\frac12)\) \(\approx\) \(0.9883243916\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.847 - 1.13i)T \)
3 \( 1 + iT \)
5 \( 1 - iT \)
23 \( 1 + (1.84 + 4.42i)T \)
good7 \( 1 - 1.84T + 7T^{2} \)
11 \( 1 + 2.86T + 11T^{2} \)
13 \( 1 + 4.13T + 13T^{2} \)
17 \( 1 - 6.96iT - 17T^{2} \)
19 \( 1 + 6.73T + 19T^{2} \)
29 \( 1 - 9.51T + 29T^{2} \)
31 \( 1 - 0.644iT - 31T^{2} \)
37 \( 1 - 10.0iT - 37T^{2} \)
41 \( 1 + 6.84T + 41T^{2} \)
43 \( 1 - 0.868T + 43T^{2} \)
47 \( 1 + 4.69iT - 47T^{2} \)
53 \( 1 - 3.26iT - 53T^{2} \)
59 \( 1 - 12.2iT - 59T^{2} \)
61 \( 1 - 7.83iT - 61T^{2} \)
67 \( 1 + 5.89T + 67T^{2} \)
71 \( 1 + 12.6iT - 71T^{2} \)
73 \( 1 + 0.824T + 73T^{2} \)
79 \( 1 - 4.81T + 79T^{2} \)
83 \( 1 - 7.08T + 83T^{2} \)
89 \( 1 + 3.47iT - 89T^{2} \)
97 \( 1 + 6.83iT - 97T^{2} \)
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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

−10.23194544018218824784156044655, −8.563454826696050164192217998939, −8.310647577464286630854796922521, −7.53399222022805295665381114465, −6.59157294086324855141427936735, −6.11665566753410021914296495154, −4.94607363484225876559833066553, −4.33635589108539124852546453094, −2.95364853393622663276838916088, −2.06904546430377785716421889350, 0.29856871924368671535893635181, 2.05253367944189166894192193543, 2.86548323081049960735004184949, 4.13554974354183115400215127102, 5.01859119389565772532261776418, 5.16671892833298508958261762868, 6.49251560734254526799830733165, 7.66408017376604558221666468415, 8.551833753190237744934184819835, 9.455907958069606565998553498835

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