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Properties

Label 2-378-21.2-c2-0-11
Degree $2$
Conductor $378$
Sign $0.0692 + 0.997i$
Analytic cond. $10.2997$
Root an. cond. $3.20932$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + (−4.33 − 2.50i)5-s + (0.0413 + 6.99i)7-s − 2.82i·8-s + (3.54 + 6.13i)10-s + (1.32 − 0.765i)11-s + 0.917·13-s + (4.89 − 8.60i)14-s + (−2.00 + 3.46i)16-s + (14.3 − 8.27i)17-s + (6.5 − 11.2i)19-s − 10.0i·20-s − 2.16·22-s + (−10.3 − 5.98i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.867 − 0.500i)5-s + (0.00591 + 0.999i)7-s − 0.353i·8-s + (0.354 + 0.613i)10-s + (0.120 − 0.0696i)11-s + 0.0705·13-s + (0.349 − 0.614i)14-s + (−0.125 + 0.216i)16-s + (0.843 − 0.486i)17-s + (0.342 − 0.592i)19-s − 0.500i·20-s − 0.0984·22-s + (−0.450 − 0.260i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0692 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0692 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.0692 + 0.997i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ 0.0692 + 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.636051 - 0.593440i\)
\(L(\frac12)\) \(\approx\) \(0.636051 - 0.593440i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 + (-0.0413 - 6.99i)T \)
good5 \( 1 + (4.33 + 2.50i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-1.32 + 0.765i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 0.917T + 169T^{2} \)
17 \( 1 + (-14.3 + 8.27i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.5 + 11.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (10.3 + 5.98i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 33.5iT - 841T^{2} \)
31 \( 1 + (-8.66 - 15.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-27.8 + 48.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 6.53iT - 1.68e3T^{2} \)
43 \( 1 - 32.7T + 1.84e3T^{2} \)
47 \( 1 + (46.3 + 26.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-72.4 + 41.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (41.4 - 23.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-40.2 + 69.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-1.21 - 2.09i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 83.3iT - 5.04e3T^{2} \)
73 \( 1 + (36.0 + 62.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-61.1 + 105. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 158. iT - 6.88e3T^{2} \)
89 \( 1 + (-46.9 - 27.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 12.0T + 9.40e3T^{2} \)
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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

−11.07175352574735399617738367651, −9.856876360613656727724002327724, −9.075102174171332178364569286451, −8.237253107380247472849585958371, −7.51992281938146080073629304819, −6.16092419866453511342834478828, −4.93886729128185748699502097489, −3.66095132006223950084358246157, −2.35161684558998687704362868121, −0.55569301181051783765251402220, 1.20761683556591696753573769594, 3.25653057356438277636495558937, 4.28825778963463645463998202920, 5.78986243868248904119060231327, 6.94107694609106313131721810279, 7.64246869884868346046885521157, 8.314396725453584481631680486701, 9.678019382310230409419023310194, 10.36676967082543691235600606720, 11.21820738858356954416503214214

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