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Properties

Label 2-414-9.4-c1-0-7
Degree $2$
Conductor $414$
Sign $-0.173 - 0.984i$
Analytic cond. $3.30580$
Root an. cond. $1.81818$
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  + (0.5 + 0.866i)2-s + (1.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−2 + 3.46i)5-s + (1.5 + 0.866i)6-s − 0.999·8-s + (1.5 − 2.59i)9-s − 3.99·10-s + (1.5 + 2.59i)11-s + 1.73i·12-s + (−2 + 3.46i)13-s + 6.92i·15-s + (−0.5 − 0.866i)16-s + 7·17-s + 3·18-s − 5·19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.866 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.894 + 1.54i)5-s + (0.612 + 0.353i)6-s − 0.353·8-s + (0.5 − 0.866i)9-s − 1.26·10-s + (0.452 + 0.783i)11-s + 0.499i·12-s + (−0.554 + 0.960i)13-s + 1.78i·15-s + (−0.125 − 0.216i)16-s + 1.69·17-s + 0.707·18-s − 1.14·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15034 + 1.37093i\)
\(L(\frac12)\) \(\approx\) \(1.15034 + 1.37093i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-1.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (-3.5 + 6.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5 + 8.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 + (6.5 - 11.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 9T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2 + 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.89694088610312784931724861751, −10.47754533640344624220151454168, −9.579008523474239910321945895614, −8.402506898665240926891753789906, −7.42319122245833173013012470687, −7.09549505499549925220111423174, −6.18509693702628057546263591449, −4.30408529317748029917261864099, −3.50853392532970637887914749328, −2.34434885812628777657352552928, 1.03182151745784563105900605671, 2.92728258201818817031841467057, 3.99111344998293195475884481248, 4.73411107136785871359875137962, 5.78018470029439707654948468699, 7.85062383936103664169981478351, 8.223085702316691053359629685518, 9.195275006362543819238279413250, 9.961474013929332202881923132184, 11.00887440719676466154542199792

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