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Properties

Label 2-1380-115.68-c1-0-7
Degree $2$
Conductor $1380$
Sign $0.266 - 0.963i$
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  + (0.707 + 0.707i)3-s + (−0.951 − 2.02i)5-s + (3.48 + 3.48i)7-s + 1.00i·9-s + 2.73i·11-s + (−0.367 − 0.367i)13-s + (0.757 − 2.10i)15-s + (5.02 + 5.02i)17-s − 5.91·19-s + 4.92i·21-s + (−2.82 − 3.87i)23-s + (−3.18 + 3.85i)25-s + (−0.707 + 0.707i)27-s − 2.00i·29-s + 2.49·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.425 − 0.904i)5-s + (1.31 + 1.31i)7-s + 0.333i·9-s + 0.823i·11-s + (−0.102 − 0.102i)13-s + (0.195 − 0.543i)15-s + (1.21 + 1.21i)17-s − 1.35·19-s + 1.07i·21-s + (−0.588 − 0.808i)23-s + (−0.637 + 0.770i)25-s + (−0.136 + 0.136i)27-s − 0.371i·29-s + 0.448·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.266 - 0.963i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.914945682\)
\(L(\frac12)\) \(\approx\) \(1.914945682\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.951 + 2.02i)T \)
23 \( 1 + (2.82 + 3.87i)T \)
good7 \( 1 + (-3.48 - 3.48i)T + 7iT^{2} \)
11 \( 1 - 2.73iT - 11T^{2} \)
13 \( 1 + (0.367 + 0.367i)T + 13iT^{2} \)
17 \( 1 + (-5.02 - 5.02i)T + 17iT^{2} \)
19 \( 1 + 5.91T + 19T^{2} \)
29 \( 1 + 2.00iT - 29T^{2} \)
31 \( 1 - 2.49T + 31T^{2} \)
37 \( 1 + (-6.38 - 6.38i)T + 37iT^{2} \)
41 \( 1 + 6.05T + 41T^{2} \)
43 \( 1 + (2.07 - 2.07i)T - 43iT^{2} \)
47 \( 1 + (-4.01 + 4.01i)T - 47iT^{2} \)
53 \( 1 + (0.794 - 0.794i)T - 53iT^{2} \)
59 \( 1 + 5.00iT - 59T^{2} \)
61 \( 1 - 9.72iT - 61T^{2} \)
67 \( 1 + (-2.09 - 2.09i)T + 67iT^{2} \)
71 \( 1 - 9.35T + 71T^{2} \)
73 \( 1 + (-8.18 - 8.18i)T + 73iT^{2} \)
79 \( 1 + 2.90T + 79T^{2} \)
83 \( 1 + (-7.48 + 7.48i)T - 83iT^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 + (-0.793 - 0.793i)T + 97iT^{2} \)
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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

−9.672361175121095081910095631582, −8.668196307899389836037637907666, −8.264800925297666688937077994658, −7.82599807802188254482055328592, −6.29301683351493282768806121602, −5.35726020195872495231154394696, −4.66425063975299076625162569900, −3.96615862362208180214389625590, −2.43762838596819645935662877914, −1.57742001589117053855494695765, 0.77941744076662899404207368060, 2.12552707053386970416908504087, 3.35920980923606009287177845305, 4.07224727746723450241446131742, 5.14838919049798834448554722255, 6.34925262916133578516781167549, 7.18900755182028627634389055435, 7.82848377748323813857748734579, 8.204203472867435302761988512482, 9.436533584609833772827698555589

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