Properties

Label 2-390-5.4-c1-0-6
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 − 2i·7-s + i·8-s − 9-s + (−1 − 2i)10-s + 2·11-s i·12-s i·13-s − 2·14-s + (1 + 2i)15-s + 16-s − 2i·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 − 0.755i·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 − 0.534·14-s + (0.258 + 0.516i)15-s + 0.250·16-s − 0.485i·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

\(L(1)\) \(\approx\) \(1.28423 - 0.793703i\)
\(L(\frac12)\) \(\approx\) \(1.28423 - 0.793703i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 - 16iT - 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−11.03988785741767423611475136287, −10.15182428810637106566355532472, −9.559245089767598976866079494335, −8.794398656612434567960187662607, −7.52845205699816026568328387423, −6.14513308344513577840423000173, −5.07192883066494602073126526013, −4.13125857353407705589556702147, −2.85808539939422729727876459943, −1.18946337852103836515029025778, 1.74012033866586859306908382209, 3.21911290967116870543334881841, 4.96935059804846759672691954704, 6.00873995796246908305735998507, 6.57053871516389154097696472168, 7.61178114907466501403572635929, 8.720742915198779871262138883141, 9.428101240143841676202585748156, 10.40349342832384002536881095366, 11.67929741531403913953414891160

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