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Properties

Label 2-390-39.32-c1-0-6
Degree $2$
Conductor $390$
Sign $0.449 - 0.893i$
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  + (0.965 − 0.258i)2-s + (−0.192 + 1.72i)3-s + (0.866 − 0.499i)4-s + (0.707 + 0.707i)5-s + (0.260 + 1.71i)6-s + (−0.213 + 0.796i)7-s + (0.707 − 0.707i)8-s + (−2.92 − 0.661i)9-s + (0.866 + 0.500i)10-s + (0.981 + 3.66i)11-s + (0.694 + 1.58i)12-s + (3.42 + 1.12i)13-s + 0.825i·14-s + (−1.35 + 1.08i)15-s + (0.500 − 0.866i)16-s + (−1.49 − 2.59i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.110 + 0.993i)3-s + (0.433 − 0.249i)4-s + (0.316 + 0.316i)5-s + (0.106 + 0.699i)6-s + (−0.0807 + 0.301i)7-s + (0.249 − 0.249i)8-s + (−0.975 − 0.220i)9-s + (0.273 + 0.158i)10-s + (0.295 + 1.10i)11-s + (0.200 + 0.458i)12-s + (0.950 + 0.310i)13-s + 0.220i·14-s + (−0.349 + 0.279i)15-s + (0.125 − 0.216i)16-s + (−0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.449 - 0.893i$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{390} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 0.449 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70607 + 1.05190i\)
\(L(\frac12)\) \(\approx\) \(1.70607 + 1.05190i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (0.192 - 1.72i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (-3.42 - 1.12i)T \)
good7 \( 1 + (0.213 - 0.796i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.981 - 3.66i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.49 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.17 - 0.314i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.76 - 3.05i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.48 + 3.74i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.00278 + 0.00278i)T - 31iT^{2} \)
37 \( 1 + (-10.9 + 2.94i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.61 - 1.23i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.40 + 1.96i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.01 + 7.01i)T - 47iT^{2} \)
53 \( 1 + 4.90iT - 53T^{2} \)
59 \( 1 + (4.67 + 1.25i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (6.87 + 11.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.00 + 11.2i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.52 - 5.70i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (8.49 + 8.49i)T + 73iT^{2} \)
79 \( 1 - 5.90T + 79T^{2} \)
83 \( 1 + (-5.36 - 5.36i)T + 83iT^{2} \)
89 \( 1 + (1.85 + 6.93i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.65 + 2.05i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.43718973486519371018246881696, −10.70462682465100615504915349826, −9.643003648843829406609585111026, −9.173202721186315300375587270594, −7.63597904494377839299212642588, −6.39177975422617485537848200799, −5.56924107201496115859136204410, −4.47045654216059081464321616929, −3.56842381233716941456316005780, −2.19632317238708268666874387783, 1.24631260138223952857670096381, 2.86632902982018801021495763928, 4.14690183201038230106156693076, 5.76771021619575528906938035558, 6.09082015535690452201718572012, 7.25595948385394692765931286107, 8.260545396167649428700732311173, 9.004961685724890954671640377731, 10.65034314824367636665350027812, 11.27578122685917110364851972766

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