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Properties

Label 2-378-189.101-c1-0-7
Degree $2$
Conductor $378$
Sign $-0.689 - 0.724i$
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 + (0.930 + 1.46i)3-s + (−0.173 − 0.984i)4-s + (−0.879 + 0.737i)5-s + (−1.71 − 0.226i)6-s + (2.51 + 0.811i)7-s + (0.866 + 0.500i)8-s + (−1.26 + 2.71i)9-s − 1.14i·10-s + (−2.40 + 2.86i)11-s + (1.27 − 1.17i)12-s + (0.794 − 2.18i)13-s + (−2.24 + 1.40i)14-s + (−1.89 − 0.597i)15-s + (−0.939 + 0.342i)16-s + 1.44·17-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (0.537 + 0.843i)3-s + (−0.0868 − 0.492i)4-s + (−0.393 + 0.329i)5-s + (−0.701 − 0.0922i)6-s + (0.951 + 0.306i)7-s + (0.306 + 0.176i)8-s + (−0.422 + 0.906i)9-s − 0.362i·10-s + (−0.724 + 0.863i)11-s + (0.368 − 0.337i)12-s + (0.220 − 0.605i)13-s + (−0.598 + 0.376i)14-s + (−0.489 − 0.154i)15-s + (−0.234 + 0.0855i)16-s + 0.351·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.689 - 0.724i$
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: \((2,\ 378,\ (\ :1/2),\ -0.689 - 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.462890 + 1.07944i\)
\(L(\frac12)\) \(\approx\) \(0.462890 + 1.07944i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.930 - 1.46i)T \)
7 \( 1 + (-2.51 - 0.811i)T \)
good5 \( 1 + (0.879 - 0.737i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (2.40 - 2.86i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.794 + 2.18i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 1.44T + 17T^{2} \)
19 \( 1 - 3.07iT - 19T^{2} \)
23 \( 1 + (0.580 - 1.59i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.08 + 2.98i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.31 - 0.231i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-4.40 + 7.63i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.11 + 1.13i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.16 + 6.58i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.49 - 8.47i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-3.71 - 2.14i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.67 - 0.972i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.347 - 0.0613i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-3.97 + 3.33i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-11.1 + 6.43i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.20 - 0.697i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.61 - 4.71i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-13.4 + 4.89i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 5.61T + 89T^{2} \)
97 \( 1 + (9.95 + 1.75i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.33400381448086641529710541011, −10.62270058973406786269745566835, −9.822811904943380170314645387168, −8.886179166927570895630921304255, −7.81220638199503748101480849056, −7.58650419255791906870244360588, −5.74339235511590883755207023581, −4.94435361747744168870401291479, −3.72808945290142702293140661351, −2.16841716134634532705555914638, 0.903300998274726735707786599564, 2.32301527789111354832630304883, 3.63497442120594597583909253496, 4.96089539197301876921410227881, 6.50277812126611521909423800339, 7.64771742447015748791212965961, 8.286703385067795785634988620644, 8.879054247010348815295936673223, 10.15128842310739345913134199573, 11.28946495613933810433626877828

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