Properties

Label 2-378-27.7-c1-0-12
Degree $2$
Conductor $378$
Sign $0.193 + 0.981i$
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  + (−0.939 + 0.342i)2-s + (0.344 + 1.69i)3-s + (0.766 − 0.642i)4-s + (−0.445 − 2.52i)5-s + (−0.904 − 1.47i)6-s + (−0.766 − 0.642i)7-s + (−0.500 + 0.866i)8-s + (−2.76 + 1.17i)9-s + (1.28 + 2.22i)10-s + (1.12 − 6.39i)11-s + (1.35 + 1.07i)12-s + (−5.08 − 1.84i)13-s + (0.939 + 0.342i)14-s + (4.13 − 1.62i)15-s + (0.173 − 0.984i)16-s + (−1.82 − 3.15i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.199 + 0.979i)3-s + (0.383 − 0.321i)4-s + (−0.199 − 1.12i)5-s + (−0.369 − 0.602i)6-s + (−0.289 − 0.242i)7-s + (−0.176 + 0.306i)8-s + (−0.920 + 0.390i)9-s + (0.405 + 0.702i)10-s + (0.339 − 1.92i)11-s + (0.391 + 0.311i)12-s + (−1.40 − 0.513i)13-s + (0.251 + 0.0914i)14-s + (1.06 − 0.420i)15-s + (0.0434 − 0.246i)16-s + (−0.442 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.193 + 0.981i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.193 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.488504 - 0.401770i\)
\(L(\frac12)\) \(\approx\) \(0.488504 - 0.401770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + (-0.344 - 1.69i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
good5 \( 1 + (0.445 + 2.52i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (-1.12 + 6.39i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (5.08 + 1.84i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.82 + 3.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.24 - 5.62i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.00 + 0.840i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (1.09 - 0.398i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-6.17 + 5.18i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-1.49 - 2.58i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.76 - 3.19i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.201 + 1.14i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.71 + 2.28i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 2.63T + 53T^{2} \)
59 \( 1 + (1.04 + 5.90i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.25 + 1.89i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-8.09 - 2.94i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (3.79 + 6.57i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.44 - 11.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.00 + 2.54i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (11.0 - 4.02i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (0.396 - 0.686i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.29 - 7.33i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.99897395734911477418849563800, −10.00809450639942320235702794912, −9.308687714823408814651203177510, −8.460651221342281270093106927896, −7.902386251845116060966568706201, −6.24296048586744533430576797329, −5.28887039604113394188165542119, −4.22555664836114817357498982890, −2.85528890045466209444957462256, −0.48558269928795126425010524512, 2.05480878092854150807734862835, 2.76590158137809375437633088513, 4.50331449962707898061155965006, 6.43491206168099399401008972769, 7.04066185732638567297215904970, 7.53303541757973956056252216746, 8.891100295294878618772108277839, 9.673804479934801645374880668623, 10.64755094432151317894007868823, 11.59534278319618618616789346027

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