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Properties

Label 2-390-5.4-c1-0-3
Degree $2$
Conductor $390$
Sign $-0.447 - 0.894i$
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 + i·3-s − 4-s + (2 − i)5-s − 6-s + 4i·7-s − i·8-s − 9-s + (1 + 2i)10-s + 2·11-s − i·12-s − i·13-s − 4·14-s + (1 + 2i)15-s + 16-s + 4i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.894 − 0.447i)5-s − 0.408·6-s + 1.51i·7-s − 0.353i·8-s − 0.333·9-s + (0.316 + 0.632i)10-s + 0.603·11-s − 0.288i·12-s − 0.277i·13-s − 1.06·14-s + (0.258 + 0.516i)15-s + 0.250·16-s + 0.970i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.763100 + 1.23472i\)
\(L(\frac12)\) \(\approx\) \(0.763100 + 1.23472i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (-2 + i)T \)
13 \( 1 + iT \)
good7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 - 14T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−11.70541008195990028035299899017, −10.44950589932283994661408709602, −9.475364833362792739908843742432, −8.927157072855751300511983482411, −8.199621978548240832095296968422, −6.59800822985621947604709322250, −5.69097944702842112330053025984, −5.19510855767203085831627464049, −3.72794716259632351621374002454, −2.04586530366640465688845444023, 1.06042640914004630859042630322, 2.44101128541060251824225185472, 3.78286145293792027194094329732, 4.99600095108582412245908631971, 6.52567964516979694630917506526, 7.02555179057765223695230575436, 8.323560979338947872027091728120, 9.453993206689234124581425350749, 10.23845136228507693970389687210, 10.95030909764592178474522169338

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