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Properties

Label 2-378-7.4-c3-0-21
Degree $2$
Conductor $378$
Sign $0.934 + 0.356i$
Analytic cond. $22.3027$
Root an. cond. $4.72257$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (−5.04 − 8.74i)5-s + (−5.48 + 17.6i)7-s − 7.99·8-s + (10.0 − 17.4i)10-s + (18.0 − 31.3i)11-s − 4.01·13-s + (−36.1 + 8.18i)14-s + (−8 − 13.8i)16-s + (18.9 − 32.8i)17-s + (−28.5 − 49.3i)19-s + 40.3·20-s + 72.2·22-s + (39.2 + 67.9i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.451 − 0.782i)5-s + (−0.296 + 0.955i)7-s − 0.353·8-s + (0.319 − 0.553i)10-s + (0.495 − 0.858i)11-s − 0.0857·13-s + (−0.689 + 0.156i)14-s + (−0.125 − 0.216i)16-s + (0.270 − 0.469i)17-s + (−0.344 − 0.596i)19-s + 0.451·20-s + 0.700·22-s + (0.355 + 0.615i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.934 + 0.356i$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ 0.934 + 0.356i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.717198144\)
\(L(\frac12)\) \(\approx\) \(1.717198144\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 \)
7 \( 1 + (5.48 - 17.6i)T \)
good5 \( 1 + (5.04 + 8.74i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-18.0 + 31.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 4.01T + 2.19e3T^{2} \)
17 \( 1 + (-18.9 + 32.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (28.5 + 49.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-39.2 - 67.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 215.T + 2.43e4T^{2} \)
31 \( 1 + (-91.1 + 157. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (156. + 270. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 186.T + 6.89e4T^{2} \)
43 \( 1 - 262.T + 7.95e4T^{2} \)
47 \( 1 + (68.9 + 119. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-273. + 474. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-1.86 + 3.23i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-57.5 - 99.5i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (313. - 543. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 533.T + 3.57e5T^{2} \)
73 \( 1 + (-149. + 258. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-433. - 750. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 244.T + 5.71e5T^{2} \)
89 \( 1 + (223. + 387. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.80e3T + 9.12e5T^{2} \)
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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

−11.16858546420726531490029340237, −9.643370733294033412679612362156, −8.795488365176515804259962498847, −8.258224254996995887539927616065, −7.00349533659087137977539773215, −5.96581172833132426244099449369, −5.11701796270823426617723377856, −4.01126813220167405180372543393, −2.71178653489864788094967212157, −0.60644053396057087768385755431, 1.21112973946991834012518566340, 2.83804410000250098266305236255, 3.86521482690341355801795773313, 4.73100722947816231326812642310, 6.37197370604197015764430318702, 7.04541405379573170727700537943, 8.150508234307664431065994989169, 9.459440981333429272359124974214, 10.45776610984907161020029984294, 10.73957384955884790555772197613

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