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Properties

Label 2-378-7.4-c3-0-27
Degree $2$
Conductor $378$
Sign $0.836 + 0.548i$
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 + (1.18 + 2.05i)5-s + (9.15 − 16.0i)7-s − 7.99·8-s + (−2.37 + 4.11i)10-s + (19.5 − 33.8i)11-s − 24.4·13-s + (37.0 − 0.232i)14-s + (−8 − 13.8i)16-s + (−23.8 + 41.3i)17-s + (−59.8 − 103. i)19-s − 9.49·20-s + 78.2·22-s + (−76.6 − 132. i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.106 + 0.183i)5-s + (0.494 − 0.869i)7-s − 0.353·8-s + (−0.0750 + 0.130i)10-s + (0.535 − 0.928i)11-s − 0.521·13-s + (0.707 − 0.00444i)14-s + (−0.125 − 0.216i)16-s + (−0.340 + 0.589i)17-s + (−0.722 − 1.25i)19-s − 0.106·20-s + 0.757·22-s + (−0.694 − 1.20i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.836 + 0.548i$
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: \((2,\ 378,\ (\ :3/2),\ 0.836 + 0.548i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.991744505\)
\(L(\frac12)\) \(\approx\) \(1.991744505\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.15 + 16.0i)T \)
good5 \( 1 + (-1.18 - 2.05i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-19.5 + 33.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 24.4T + 2.19e3T^{2} \)
17 \( 1 + (23.8 - 41.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (59.8 + 103. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (76.6 + 132. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 215.T + 2.43e4T^{2} \)
31 \( 1 + (-29.6 + 51.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-104. - 181. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 415.T + 6.89e4T^{2} \)
43 \( 1 + 452.T + 7.95e4T^{2} \)
47 \( 1 + (-114. - 198. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-220. + 382. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-362. + 627. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-170. - 295. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (125. - 217. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 209.T + 3.57e5T^{2} \)
73 \( 1 + (60.9 - 105. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (399. + 692. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 116.T + 5.71e5T^{2} \)
89 \( 1 + (-183. - 317. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.04e3T + 9.12e5T^{2} \)
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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

−10.87070664620955452047584107193, −10.03071447697446421723026582471, −8.654907988586221602881433691364, −8.119115488781755973144398896969, −6.78703125055906166827614544818, −6.32380321430228404765040269371, −4.80979641873658777863656725791, −4.09316699384181222764124090395, −2.60160759894985429364896137435, −0.62507684867851037213381026252, 1.50747775298019524446271504091, 2.52914059314062382503992380566, 4.04792877512697482195685866141, 5.02381148088380448739237810532, 5.96328148553921166004405769936, 7.24249462531004898651656229596, 8.431023382103376721741919817729, 9.368339295735785717512608973940, 10.07700578492884291164453992431, 11.20808695868042572506135734604

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