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Properties

Label 2-378-63.38-c1-0-3
Degree $2$
Conductor $378$
Sign $0.700 + 0.713i$
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  − i·2-s − 4-s + (0.0338 − 0.0585i)5-s + (1.19 + 2.35i)7-s + i·8-s + (−0.0585 − 0.0338i)10-s + (3.40 − 1.96i)11-s + (3.32 − 1.92i)13-s + (2.35 − 1.19i)14-s + 16-s + (−0.775 + 1.34i)17-s + (5.06 − 2.92i)19-s + (−0.0338 + 0.0585i)20-s + (−1.96 − 3.40i)22-s + (−4.78 − 2.76i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.0151 − 0.0261i)5-s + (0.452 + 0.891i)7-s + 0.353i·8-s + (−0.0185 − 0.0106i)10-s + (1.02 − 0.592i)11-s + (0.922 − 0.532i)13-s + (0.630 − 0.320i)14-s + 0.250·16-s + (−0.188 + 0.325i)17-s + (1.16 − 0.670i)19-s + (−0.00755 + 0.0130i)20-s + (−0.418 − 0.725i)22-s + (−0.998 − 0.576i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.700 + 0.713i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.700 + 0.713i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33516 - 0.559799i\)
\(L(\frac12)\) \(\approx\) \(1.33516 - 0.559799i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (-1.19 - 2.35i)T \)
good5 \( 1 + (-0.0338 + 0.0585i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.40 + 1.96i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.32 + 1.92i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.775 - 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.06 + 2.92i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.78 + 2.76i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.20 + 0.697i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.26iT - 31T^{2} \)
37 \( 1 + (4.35 + 7.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.17 - 8.96i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.735 + 1.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.54T + 47T^{2} \)
53 \( 1 + (-6.28 - 3.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 9.40T + 59T^{2} \)
61 \( 1 + 0.0815iT - 61T^{2} \)
67 \( 1 + 15.3T + 67T^{2} \)
71 \( 1 + 4.30iT - 71T^{2} \)
73 \( 1 + (-6.12 - 3.53i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 6.84T + 79T^{2} \)
83 \( 1 + (3.93 - 6.81i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.84 + 10.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.363 + 0.209i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.32323320398817434270332183672, −10.56394110484285085639835126027, −9.220900428965529388602538824868, −8.824559562046346458533454111260, −7.73357840822426117569117382012, −6.22543804636961341239655050863, −5.37597060520171867980658688289, −4.03415060795734879128841054455, −2.89205796464543101869650569969, −1.34947000286231572192764252067, 1.42077436925120134338382990794, 3.70877824563929242642637849030, 4.50376789331447374359133720214, 5.83063427331413074677683438298, 6.84429378547461374216323743946, 7.58342776863351145968631649627, 8.615714109976406739578435704727, 9.557216378114821333843739227264, 10.43343765671877669080314883697, 11.56041859203089973140008277181

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