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Properties

Label 2-378-189.88-c1-0-9
Degree $2$
Conductor $378$
Sign $0.792 - 0.610i$
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  + (−0.766 − 0.642i)2-s + (1.35 + 1.08i)3-s + (0.173 + 0.984i)4-s + (0.827 − 0.694i)5-s + (−0.340 − 1.69i)6-s + (−1.68 + 2.04i)7-s + (0.500 − 0.866i)8-s + (0.659 + 2.92i)9-s − 1.07·10-s + (0.322 + 0.270i)11-s + (−0.830 + 1.51i)12-s + (4.70 + 1.71i)13-s + (2.60 − 0.482i)14-s + (1.86 − 0.0440i)15-s + (−0.939 + 0.342i)16-s + 1.50·17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (0.780 + 0.624i)3-s + (0.0868 + 0.492i)4-s + (0.369 − 0.310i)5-s + (−0.139 − 0.693i)6-s + (−0.635 + 0.771i)7-s + (0.176 − 0.306i)8-s + (0.219 + 0.975i)9-s − 0.341·10-s + (0.0970 + 0.0814i)11-s + (−0.239 + 0.438i)12-s + (1.30 + 0.474i)13-s + (0.695 − 0.129i)14-s + (0.482 − 0.0113i)15-s + (−0.234 + 0.0855i)16-s + 0.363·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.792 - 0.610i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.792 - 0.610i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29376 + 0.440399i\)
\(L(\frac12)\) \(\approx\) \(1.29376 + 0.440399i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
3 \( 1 + (-1.35 - 1.08i)T \)
7 \( 1 + (1.68 - 2.04i)T \)
good5 \( 1 + (-0.827 + 0.694i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-0.322 - 0.270i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (-4.70 - 1.71i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 - 1.50T + 17T^{2} \)
19 \( 1 + 4.08T + 19T^{2} \)
23 \( 1 + (-0.846 - 0.308i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-6.25 + 2.27i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (0.750 + 4.25i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (5.53 - 9.58i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.75 - 2.09i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.74 + 9.90i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.31 + 7.43i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-3.58 + 6.21i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.57 + 1.29i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.666 - 3.77i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-2.97 + 2.49i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (5.08 + 8.80i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.68 - 2.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.363 - 0.305i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (8.32 - 3.03i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + (-1.08 + 6.16i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.28229325278098574927228696824, −10.30013869169038015554645259333, −9.572659985392087450613125569015, −8.768345866886458862323637578004, −8.316578214759721548391724771862, −6.80599545483975795759741103976, −5.61301146942145568241125129953, −4.17310749644784682922312349161, −3.11491192920196141916436244301, −1.86543539637485873459837801049, 1.13024271255560865020770720456, 2.81411472256941505021213606831, 4.04583586177688285726801086969, 6.00638215002977006580606163557, 6.60288106315432930993385575734, 7.54850898439755209325550339666, 8.470079660280247331390074840725, 9.220889849719256406713278619884, 10.29882971850980009419941768033, 10.87751359331539420868122362419

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