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Properties

Label 2-378-189.101-c1-0-8
Degree $2$
Conductor $378$
Sign $0.806 + 0.591i$
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  + (−0.642 + 0.766i)2-s + (−1.70 + 0.281i)3-s + (−0.173 − 0.984i)4-s + (−3.39 + 2.85i)5-s + (0.883 − 1.48i)6-s + (−2.33 − 1.24i)7-s + (0.866 + 0.500i)8-s + (2.84 − 0.960i)9-s − 4.43i·10-s + (−1.47 + 1.75i)11-s + (0.573 + 1.63i)12-s + (1.38 − 3.80i)13-s + (2.45 − 0.983i)14-s + (5.00 − 5.83i)15-s + (−0.939 + 0.342i)16-s + 3.73·17-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (−0.986 + 0.162i)3-s + (−0.0868 − 0.492i)4-s + (−1.52 + 1.27i)5-s + (0.360 − 0.608i)6-s + (−0.881 − 0.472i)7-s + (0.306 + 0.176i)8-s + (0.947 − 0.320i)9-s − 1.40i·10-s + (−0.444 + 0.529i)11-s + (0.165 + 0.471i)12-s + (0.384 − 1.05i)13-s + (0.656 − 0.262i)14-s + (1.29 − 1.50i)15-s + (−0.234 + 0.0855i)16-s + 0.905·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.806 + 0.591i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.806 + 0.591i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.252624 - 0.0826532i\)
\(L(\frac12)\) \(\approx\) \(0.252624 - 0.0826532i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 - 0.766i)T \)
3 \( 1 + (1.70 - 0.281i)T \)
7 \( 1 + (2.33 + 1.24i)T \)
good5 \( 1 + (3.39 - 2.85i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (1.47 - 1.75i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-1.38 + 3.80i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 3.73T + 17T^{2} \)
19 \( 1 - 6.57iT - 19T^{2} \)
23 \( 1 + (-1.40 + 3.85i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.81 + 4.98i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (0.0937 - 0.0165i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.45 + 2.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.325 - 0.118i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.361 + 2.04i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.394 + 2.23i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-1.50 - 0.868i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (11.4 + 4.17i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (7.35 + 1.29i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-3.42 + 2.87i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (3.12 - 1.80i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-11.7 + 6.76i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.55 + 1.30i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.970 - 0.353i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + 6.26T + 89T^{2} \)
97 \( 1 + (10.4 + 1.84i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.93794301590829068227193564851, −10.44881963189131648929683560159, −9.835565909687872486732318735698, −7.983900967059238352482055206689, −7.52832115688605087664284516328, −6.60717581639387979063815826168, −5.78740172765454938263241188941, −4.23193390833020296902899357797, −3.27984210996731948664783894069, −0.29851005275420215604405839818, 1.05999633751738592611024949942, 3.35363522692190260356288000165, 4.48917856375112907017266952917, 5.45983913299835363550880594560, 6.89654878755846349840673900332, 7.79136982321546299183140197096, 8.876557754069208759322056962623, 9.465749962125045719743609124902, 10.91715134653253013629400050968, 11.50861364379025540439534666661

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