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Properties

Label 2-378-9.7-c1-0-5
Degree $2$
Conductor $378$
Sign $-0.939 + 0.342i$
Analytic cond. $3.01834$
Root an. cond. $1.73733$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.5 − 2.59i)5-s + (−0.5 + 0.866i)7-s + 0.999·8-s + 3·10-s + (−3 + 5.19i)11-s + (−1 − 1.73i)13-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 6·17-s − 7·19-s + (−1.50 + 2.59i)20-s + (−3 − 5.19i)22-s + (1.5 + 2.59i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.670 − 1.16i)5-s + (−0.188 + 0.327i)7-s + 0.353·8-s + 0.948·10-s + (−0.904 + 1.56i)11-s + (−0.277 − 0.480i)13-s + (−0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s − 1.45·17-s − 1.60·19-s + (−0.335 + 0.580i)20-s + (−0.639 − 1.10i)22-s + (0.312 + 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.82172178761459126534247229491, −9.833794920243197303086819951413, −8.943575628321349450974308630359, −8.172850567326801726922966235812, −7.37744561624587485312104480999, −6.22272127375016884944758152832, −4.86655794452651834681308798853, −4.40168990874711540339157466811, −2.19438800668068560131758197913, 0, 2.48335868333655857253174963387, 3.41515839077614286366190499300, 4.57197135055793564194285155868, 6.31597036463547930931719513459, 7.08339702267189722087157721419, 8.250330189856184009945970807373, 8.905452988034738469766559974726, 10.37963010064293986895324906345, 10.90450564140489019519298308146

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