Properties

Label 2-378-189.41-c1-0-15
Degree $2$
Conductor $378$
Sign $0.661 + 0.749i$
Analytic cond. $3.01834$
Root an. cond. $1.73733$
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  + (0.984 − 0.173i)2-s + (−1.47 + 0.900i)3-s + (0.939 − 0.342i)4-s + (−1.23 − 1.03i)5-s + (−1.30 + 1.14i)6-s + (1.54 − 2.14i)7-s + (0.866 − 0.5i)8-s + (1.37 − 2.66i)9-s + (−1.39 − 0.805i)10-s + (−0.858 − 1.02i)11-s + (−1.08 + 1.35i)12-s + (3.00 + 0.530i)13-s + (1.15 − 2.38i)14-s + (2.75 + 0.421i)15-s + (0.766 − 0.642i)16-s + (1.40 − 2.43i)17-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (−0.854 + 0.519i)3-s + (0.469 − 0.171i)4-s + (−0.551 − 0.463i)5-s + (−0.531 + 0.466i)6-s + (0.585 − 0.810i)7-s + (0.306 − 0.176i)8-s + (0.459 − 0.888i)9-s + (−0.441 − 0.254i)10-s + (−0.258 − 0.308i)11-s + (−0.312 + 0.390i)12-s + (0.834 + 0.147i)13-s + (0.308 − 0.636i)14-s + (0.712 + 0.108i)15-s + (0.191 − 0.160i)16-s + (0.341 − 0.591i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.661 + 0.749i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.661 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37619 - 0.620709i\)
\(L(\frac12)\) \(\approx\) \(1.37619 - 0.620709i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 + 0.173i)T \)
3 \( 1 + (1.47 - 0.900i)T \)
7 \( 1 + (-1.54 + 2.14i)T \)
good5 \( 1 + (1.23 + 1.03i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (0.858 + 1.02i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-3.00 - 0.530i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-1.40 + 2.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.55 + 1.47i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.215 + 0.591i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.585 - 0.103i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.88 - 5.18i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (-4.27 + 7.41i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.515 - 2.92i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (8.07 - 6.77i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (7.18 + 2.61i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 9.51iT - 53T^{2} \)
59 \( 1 + (4.31 + 3.62i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.62 - 4.47i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.70 - 9.64i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.73 - 3.31i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.98 + 4.61i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.490 - 2.78i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.953 - 5.41i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (5.41 + 9.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.0 - 13.2i)T + (-16.8 + 95.5i)T^{2} \)
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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

−11.32887021315580879572562916956, −10.67625942451098803997164490358, −9.681980549014437931104919292301, −8.375297031025388087585125666365, −7.30948873991655029257189711892, −6.23481756213765103143663698063, −5.10000905728013110797281960339, −4.41490813800781094676934079893, −3.40362503917120653936326588728, −1.01249444521455458243336629754, 1.80278057936429505728183909519, 3.41286955972874482107624600035, 4.79966923606194057350400497073, 5.69725368627196754115741263291, 6.52059565174989111026836939328, 7.65389617429929400326023663516, 8.270280043742600211565704452962, 9.945105384000268464790703997325, 11.06390897745244621988074472364, 11.54141624116800090169827795870

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