Properties

Label 2-378-27.7-c1-0-6
Degree $2$
Conductor $378$
Sign $0.851 + 0.524i$
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 + (−1.73 − 0.0833i)3-s + (0.766 − 0.642i)4-s + (0.111 + 0.629i)5-s + (1.65 − 0.513i)6-s + (−0.766 − 0.642i)7-s + (−0.500 + 0.866i)8-s + (2.98 + 0.288i)9-s + (−0.319 − 0.553i)10-s + (0.0504 − 0.286i)11-s + (−1.37 + 1.04i)12-s + (−4.81 − 1.75i)13-s + (0.939 + 0.342i)14-s + (−0.139 − 1.09i)15-s + (0.173 − 0.984i)16-s + (2.78 + 4.82i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (−0.998 − 0.0481i)3-s + (0.383 − 0.321i)4-s + (0.0496 + 0.281i)5-s + (0.675 − 0.209i)6-s + (−0.289 − 0.242i)7-s + (−0.176 + 0.306i)8-s + (0.995 + 0.0961i)9-s + (−0.101 − 0.175i)10-s + (0.0152 − 0.0862i)11-s + (−0.398 + 0.302i)12-s + (−1.33 − 0.486i)13-s + (0.251 + 0.0914i)14-s + (−0.0360 − 0.283i)15-s + (0.0434 − 0.246i)16-s + (0.676 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.851 + 0.524i$
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: \((2,\ 378,\ (\ :1/2),\ 0.851 + 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.625012 - 0.176896i\)
\(L(\frac12)\) \(\approx\) \(0.625012 - 0.176896i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 0.0833i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
good5 \( 1 + (-0.111 - 0.629i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (-0.0504 + 0.286i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (4.81 + 1.75i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.78 - 4.82i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.87 + 6.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.72 + 3.12i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-4.43 + 1.61i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-5.53 + 4.64i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-0.949 - 1.64i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.68 + 2.79i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.947 + 5.37i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-4.88 - 4.10i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 8.29T + 53T^{2} \)
59 \( 1 + (1.86 + 10.5i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-3.92 - 3.29i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (7.99 + 2.90i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-1.99 - 3.46i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.00 - 1.74i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.76 - 3.55i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (9.78 - 3.56i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (8.78 - 15.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.49 + 14.1i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.10092287024071829441550318324, −10.27318132902574916789312757949, −9.775869405901879677298499800663, −8.460455138491702246822909704416, −7.27166295686422025112787334199, −6.73504972996304359072311297434, −5.60222268464051456084264938159, −4.61042830427344790308427068661, −2.75421156188430795729490809205, −0.73342619521758851169150738651, 1.21725097752174813599695414508, 3.04422688392116672675078863323, 4.73446554000394037466465366683, 5.58518118174448025521442923013, 6.90786318126285289935973634821, 7.52022462647989584522917889046, 8.923943082227586533462097347969, 9.876506118125002310089959041569, 10.26013462059533615315391307637, 11.71630497988240693701848553352

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