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Properties

Label 2-390-195.89-c1-0-2
Degree $2$
Conductor $390$
Sign $0.0309 - 0.999i$
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  + (0.258 − 0.965i)2-s + (−1.05 − 1.37i)3-s + (−0.866 − 0.499i)4-s + (1.49 + 1.66i)5-s + (−1.59 + 0.667i)6-s + (−4.36 + 1.16i)7-s + (−0.707 + 0.707i)8-s + (−0.758 + 2.90i)9-s + (1.99 − 1.01i)10-s + (−1.26 + 4.71i)11-s + (0.231 + 1.71i)12-s + (−2.81 − 2.25i)13-s + 4.51i·14-s + (0.692 − 3.81i)15-s + (0.500 + 0.866i)16-s + (−2.59 − 1.49i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.611 − 0.791i)3-s + (−0.433 − 0.249i)4-s + (0.669 + 0.742i)5-s + (−0.652 + 0.272i)6-s + (−1.64 + 0.441i)7-s + (−0.249 + 0.249i)8-s + (−0.252 + 0.967i)9-s + (0.629 − 0.321i)10-s + (−0.381 + 1.42i)11-s + (0.0667 + 0.495i)12-s + (−0.780 − 0.624i)13-s + 1.20i·14-s + (0.178 − 0.983i)15-s + (0.125 + 0.216i)16-s + (−0.629 − 0.363i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0309 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0309 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.0309 - 0.999i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.0309 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.233644 + 0.226533i\)
\(L(\frac12)\) \(\approx\) \(0.233644 + 0.226533i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (1.05 + 1.37i)T \)
5 \( 1 + (-1.49 - 1.66i)T \)
13 \( 1 + (2.81 + 2.25i)T \)
good7 \( 1 + (4.36 - 1.16i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.26 - 4.71i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.59 + 1.49i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.54 - 0.681i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.563 + 0.325i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.65 + 3.84i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.43 + 4.43i)T + 31iT^{2} \)
37 \( 1 + (2.35 - 8.79i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.58 + 0.425i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.34 + 4.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.70 - 5.70i)T - 47iT^{2} \)
53 \( 1 - 7.06T + 53T^{2} \)
59 \( 1 + (3.39 - 0.908i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.24 - 9.07i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.9 + 3.46i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.11 + 4.17i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.09 + 1.09i)T + 73iT^{2} \)
79 \( 1 + 0.529T + 79T^{2} \)
83 \( 1 + (0.150 + 0.150i)T + 83iT^{2} \)
89 \( 1 + (-2.02 + 7.53i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.09 - 11.5i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.76836089049333461862817936077, −10.36892262648850742650718294695, −10.13754569663056574216310138040, −9.137889943352302898548774549151, −7.51676177145272388290463035210, −6.64943749595406589296073065374, −5.93301281789661533786047899346, −4.75710319768491544167021489427, −2.89669670796148207475171387363, −2.19810777267828559113852266764, 0.20124356959612591537355705254, 3.16379568745668297144421870422, 4.31162695678183181295536254315, 5.39028209191190052369951475795, 6.22033282380624821947445243489, 6.88515595897144232843195298129, 8.672765065755589522597854038743, 9.176061841121714734627287680863, 10.09815744367378832850300342934, 10.83542806881195064261418119150

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