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Properties

Label 2-390-65.18-c1-0-10
Degree $2$
Conductor $390$
Sign $0.549 + 0.835i$
Analytic cond. $3.11416$
Root an. cond. $1.76469$
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  + i·2-s + (−0.707 − 0.707i)3-s − 4-s + (1.57 − 1.59i)5-s + (0.707 − 0.707i)6-s − 1.24·7-s − i·8-s + 1.00i·9-s + (1.59 + 1.57i)10-s + (−2.63 − 2.63i)11-s + (0.707 + 0.707i)12-s + (2.42 − 2.67i)13-s − 1.24i·14-s + (−2.23 + 0.0155i)15-s + 16-s + (0.658 + 0.658i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (0.702 − 0.711i)5-s + (0.288 − 0.288i)6-s − 0.469·7-s − 0.353i·8-s + 0.333i·9-s + (0.503 + 0.496i)10-s + (−0.794 − 0.794i)11-s + (0.204 + 0.204i)12-s + (0.671 − 0.741i)13-s − 0.332i·14-s + (−0.577 + 0.00400i)15-s + 0.250·16-s + (0.159 + 0.159i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.549 + 0.835i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.549 + 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908548 - 0.489687i\)
\(L(\frac12)\) \(\approx\) \(0.908548 - 0.489687i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-1.57 + 1.59i)T \)
13 \( 1 + (-2.42 + 2.67i)T \)
good7 \( 1 + 1.24T + 7T^{2} \)
11 \( 1 + (2.63 + 2.63i)T + 11iT^{2} \)
17 \( 1 + (-0.658 - 0.658i)T + 17iT^{2} \)
19 \( 1 + (5.37 + 5.37i)T + 19iT^{2} \)
23 \( 1 + (-6.45 + 6.45i)T - 23iT^{2} \)
29 \( 1 - 8.74iT - 29T^{2} \)
31 \( 1 + (-1 + i)T - 31iT^{2} \)
37 \( 1 + 1.70T + 37T^{2} \)
41 \( 1 + (-6.86 + 6.86i)T - 41iT^{2} \)
43 \( 1 + (1.99 - 1.99i)T - 43iT^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + (-2.99 - 2.99i)T + 53iT^{2} \)
59 \( 1 + (5.35 - 5.35i)T - 59iT^{2} \)
61 \( 1 + 5.64T + 61T^{2} \)
67 \( 1 - 6.45iT - 67T^{2} \)
71 \( 1 + (8.57 - 8.57i)T - 71iT^{2} \)
73 \( 1 - 6.74iT - 73T^{2} \)
79 \( 1 + 6.92iT - 79T^{2} \)
83 \( 1 + 1.95T + 83T^{2} \)
89 \( 1 + (-7.93 + 7.93i)T - 89iT^{2} \)
97 \( 1 + 14.3iT - 97T^{2} \)
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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

−10.88156046704745199290878928427, −10.42791452694516230641396487405, −8.834066734093404486919681148903, −8.661995762012042596116328967964, −7.26252499879803766702981045470, −6.27409039049168281237913127434, −5.57458529571807636938599102658, −4.62164701127247956673013981677, −2.80958647077236745465534011252, −0.73256836690906133853819310156, 1.90753167717470840646514486063, 3.22568761715257913059125161802, 4.40408871927171618369288924081, 5.66594040493413843379472392680, 6.51016885392006659725247181475, 7.74902646467398277377156593034, 9.186754183876626026572655517447, 9.818027039804199369830415735822, 10.56714981366660225037799539301, 11.22727614512563973042039795285

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