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Properties

Label 2-378-63.25-c1-0-2
Degree $2$
Conductor $378$
Sign $0.213 + 0.976i$
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  − 2-s + 4-s + (−0.230 + 0.398i)5-s + (0.0665 − 2.64i)7-s − 8-s + (0.230 − 0.398i)10-s + (−1.82 − 3.15i)11-s + (0.730 + 1.26i)13-s + (−0.0665 + 2.64i)14-s + 16-s + (1.86 − 3.23i)17-s + (−2.02 − 3.51i)19-s + (−0.230 + 0.398i)20-s + (1.82 + 3.15i)22-s + (0.566 − 0.981i)23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (−0.102 + 0.178i)5-s + (0.0251 − 0.999i)7-s − 0.353·8-s + (0.0728 − 0.126i)10-s + (−0.549 − 0.952i)11-s + (0.202 + 0.350i)13-s + (−0.0177 + 0.706i)14-s + 0.250·16-s + (0.452 − 0.784i)17-s + (−0.465 − 0.805i)19-s + (−0.0514 + 0.0891i)20-s + (0.388 + 0.673i)22-s + (0.118 − 0.204i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.213 + 0.976i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.213 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.649574 - 0.523049i\)
\(L(\frac12)\) \(\approx\) \(0.649574 - 0.523049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 + (-0.0665 + 2.64i)T \)
good5 \( 1 + (0.230 - 0.398i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.82 + 3.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.730 - 1.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.86 + 3.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.02 + 3.51i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.566 + 0.981i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.48 + 7.77i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.514T + 31T^{2} \)
37 \( 1 + (4.55 + 7.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.472 - 0.819i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.66 + 8.07i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.32T + 47T^{2} \)
53 \( 1 + (6.21 - 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + 2.32T + 67T^{2} \)
71 \( 1 + 1.67T + 71T^{2} \)
73 \( 1 + (6.62 - 11.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.00T + 79T^{2} \)
83 \( 1 + (3.32 - 5.75i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.36 - 2.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.59 - 9.68i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.98099357618831381422807349460, −10.36613413317489379430033459095, −9.315975535678588692968083896487, −8.399000974653278118712031051689, −7.44180195446870684077644826272, −6.70094082835537741282779104536, −5.42014497308620620823936540920, −3.98063074580587535247293918254, −2.67522571548593585623230381822, −0.71931949668298495679503022046, 1.74795103451137140832114710543, 3.10764862115675558093630375663, 4.79589134688594176208177987037, 5.88147479707962295150086703306, 6.93245395939526850700686159548, 8.195296222362346998137119271447, 8.572580176875748798472565903100, 9.849690213118581188355831474526, 10.39096459523702171680004694638, 11.52250869025758484807946442601

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