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Properties

Label 2-378-63.34-c2-0-5
Degree $2$
Conductor $378$
Sign $0.202 - 0.979i$
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  + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (−2.08 + 1.20i)5-s + (−3.20 + 6.22i)7-s − 2.82·8-s + 3.40i·10-s + (6.79 − 11.7i)11-s + (−11.7 + 6.78i)13-s + (5.34 + 8.32i)14-s + (−2.00 + 3.46i)16-s + 19.8i·17-s + 34.4i·19-s + (4.16 + 2.40i)20-s + (−9.61 − 16.6i)22-s + (−4.33 − 7.50i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.416 + 0.240i)5-s + (−0.458 + 0.888i)7-s − 0.353·8-s + 0.340i·10-s + (0.618 − 1.07i)11-s + (−0.904 + 0.522i)13-s + (0.382 + 0.594i)14-s + (−0.125 + 0.216i)16-s + 1.16i·17-s + 1.81i·19-s + (0.208 + 0.120i)20-s + (−0.437 − 0.757i)22-s + (−0.188 − 0.326i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.202 - 0.979i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ 0.202 - 0.979i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.777977 + 0.633656i\)
\(L(\frac12)\) \(\approx\) \(0.777977 + 0.633656i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 \)
7 \( 1 + (3.20 - 6.22i)T \)
good5 \( 1 + (2.08 - 1.20i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-6.79 + 11.7i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (11.7 - 6.78i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 19.8iT - 289T^{2} \)
19 \( 1 - 34.4iT - 361T^{2} \)
23 \( 1 + (4.33 + 7.50i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (12.9 - 22.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (21.6 - 12.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 39.6T + 1.36e3T^{2} \)
41 \( 1 + (-41.8 + 24.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-13.1 + 22.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-14.5 - 8.41i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 35.6T + 2.80e3T^{2} \)
59 \( 1 + (63.9 - 36.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (78.5 + 45.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (10.3 + 17.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 38.1T + 5.04e3T^{2} \)
73 \( 1 + 12.0iT - 5.32e3T^{2} \)
79 \( 1 + (65.4 - 113. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-27.6 - 15.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 118. iT - 7.92e3T^{2} \)
97 \( 1 + (-99.4 - 57.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.41396709040865769383737018470, −10.62341604702762845407142197022, −9.544448418403170953667086272821, −8.787750581132885203481521737161, −7.68530229368788154110992070845, −6.25810318205393784094949030958, −5.62151509913032692075015138282, −4.05998660773889048946213628672, −3.23223981881227328358273431864, −1.78641879353286142893522440073, 0.38909600596730931237063772714, 2.74234189187048478033622058580, 4.23547206663920844234127534944, 4.80639834660394694743433196761, 6.28135060249466901656201746026, 7.35694171501135978956665722799, 7.63450115163180119691041360688, 9.295496543764473763453972282695, 9.706220284408803117104586319812, 11.09774601195867864412508831407

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