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Properties

Label 2-1380-92.91-c1-0-56
Degree $2$
Conductor $1380$
Sign $0.988 - 0.149i$
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  + (1.40 + 0.178i)2-s − i·3-s + (1.93 + 0.500i)4-s + i·5-s + (0.178 − 1.40i)6-s − 1.14·7-s + (2.62 + 1.04i)8-s − 9-s + (−0.178 + 1.40i)10-s + 0.631·11-s + (0.500 − 1.93i)12-s + 4.17·13-s + (−1.60 − 0.203i)14-s + 15-s + (3.49 + 1.93i)16-s + 2.21i·17-s + ⋯
L(s)  = 1  + (0.992 + 0.126i)2-s − 0.577i·3-s + (0.968 + 0.250i)4-s + 0.447i·5-s + (0.0728 − 0.572i)6-s − 0.432·7-s + (0.928 + 0.370i)8-s − 0.333·9-s + (−0.0564 + 0.443i)10-s + 0.190·11-s + (0.144 − 0.558i)12-s + 1.15·13-s + (−0.428 − 0.0545i)14-s + 0.258·15-s + (0.874 + 0.484i)16-s + 0.537i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.988 - 0.149i$
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: \((2,\ 1380,\ (\ :1/2),\ 0.988 - 0.149i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.337390004\)
\(L(\frac12)\) \(\approx\) \(3.337390004\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.178i)T \)
3 \( 1 + iT \)
5 \( 1 - iT \)
23 \( 1 + (4.41 - 1.87i)T \)
good7 \( 1 + 1.14T + 7T^{2} \)
11 \( 1 - 0.631T + 11T^{2} \)
13 \( 1 - 4.17T + 13T^{2} \)
17 \( 1 - 2.21iT - 17T^{2} \)
19 \( 1 - 4.14T + 19T^{2} \)
29 \( 1 - 7.71T + 29T^{2} \)
31 \( 1 + 7.70iT - 31T^{2} \)
37 \( 1 - 2.64iT - 37T^{2} \)
41 \( 1 + 1.13T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 - 3.24iT - 47T^{2} \)
53 \( 1 + 7.41iT - 53T^{2} \)
59 \( 1 - 0.725iT - 59T^{2} \)
61 \( 1 - 9.63iT - 61T^{2} \)
67 \( 1 + 1.72T + 67T^{2} \)
71 \( 1 + 1.79iT - 71T^{2} \)
73 \( 1 + 2.76T + 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 3.95T + 83T^{2} \)
89 \( 1 + 11.7iT - 89T^{2} \)
97 \( 1 + 9.36iT - 97T^{2} \)
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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

−9.755652971270907909181068422660, −8.514967348499572945112020710795, −7.77158152644021860652455777843, −6.97462675944327050803449054488, −6.11142667684715299507330846102, −5.79820820024111447030456595253, −4.37719488391768351352617384015, −3.51753671219320667562104205808, −2.65245939178265414180314323478, −1.38824047626089626824908097825, 1.19813621581189193250506587091, 2.74257713785540530033215140948, 3.60491492356665162379678503938, 4.41594286444040316908026437004, 5.27743604109010009973925312067, 6.06913407192586275688662621454, 6.82843241786809444975918325975, 7.928093171641067693467502558725, 8.833702068689373301422423653926, 9.700513832923555571838802986133

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