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Properties

Label 2-378-21.5-c3-0-25
Degree $2$
Conductor $378$
Sign $-0.516 + 0.856i$
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  + (−1.73 − i)2-s + (1.99 + 3.46i)4-s + (3.80 − 6.59i)5-s + (18.4 − 1.98i)7-s − 7.99i·8-s + (−13.1 + 7.61i)10-s + (−35.9 + 20.7i)11-s − 69.6i·13-s + (−33.8 − 14.9i)14-s + (−8 + 13.8i)16-s + (−54.3 − 94.1i)17-s + (82.5 + 47.6i)19-s + 30.4·20-s + 82.9·22-s + (109. + 63.3i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.340 − 0.589i)5-s + (0.994 − 0.106i)7-s − 0.353i·8-s + (−0.416 + 0.240i)10-s + (−0.984 + 0.568i)11-s − 1.48i·13-s + (−0.646 − 0.286i)14-s + (−0.125 + 0.216i)16-s + (−0.775 − 1.34i)17-s + (0.996 + 0.575i)19-s + 0.340·20-s + 0.804·22-s + (0.994 + 0.574i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 + 0.856i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.516 + 0.856i$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ -0.516 + 0.856i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.254250543\)
\(L(\frac12)\) \(\approx\) \(1.254250543\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 + i)T \)
3 \( 1 \)
7 \( 1 + (-18.4 + 1.98i)T \)
good5 \( 1 + (-3.80 + 6.59i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (35.9 - 20.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 69.6iT - 2.19e3T^{2} \)
17 \( 1 + (54.3 + 94.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-82.5 - 47.6i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-109. - 63.3i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 208. iT - 2.43e4T^{2} \)
31 \( 1 + (243. - 140. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-113. + 195. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 446.T + 6.89e4T^{2} \)
43 \( 1 - 228.T + 7.95e4T^{2} \)
47 \( 1 + (-139. + 241. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-118. + 68.1i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (405. + 702. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (489. + 282. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (475. + 822. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 448. iT - 3.57e5T^{2} \)
73 \( 1 + (-267. + 154. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-139. + 242. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 495.T + 5.71e5T^{2} \)
89 \( 1 + (-248. + 429. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 389. iT - 9.12e5T^{2} \)
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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

−10.64476861723618457485167627540, −9.689832095207509041400665297399, −8.888885773067389096624068855365, −7.79189682782736836108072648124, −7.33737800695208944457921724818, −5.44001496163222531011349454609, −4.92030821670038937524024209656, −3.16093204890301530239685398091, −1.86476273285936718940665437764, −0.53313111020394260410091503021, 1.49398075389057733659434889416, 2.70062827999491940381453895367, 4.48586296595226947837949485981, 5.58590950562176298634668059924, 6.66384685717997990962320710585, 7.48509643791010227340355573356, 8.593038353369092625270830285649, 9.159415067802177744337297083456, 10.59879040579666927531537345561, 10.88043833647928254665500098739

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