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Properties

Label 2-14-7.4-c5-0-3
Degree $2$
Conductor $14$
Sign $-0.340 + 0.940i$
Analytic cond. $2.24537$
Root an. cond. $1.49845$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 3.46i)2-s + (5.38 − 9.33i)3-s + (−7.99 + 13.8i)4-s + (−53.0 − 91.8i)5-s − 43.1·6-s + (124. + 36.3i)7-s + 63.9·8-s + (63.4 + 109. i)9-s + (−212. + 367. i)10-s + (46.7 − 80.9i)11-s + (86.2 + 149. i)12-s + 661.·13-s + (−122. − 503. i)14-s − 1.14e3·15-s + (−128 − 221. i)16-s + (−227. + 394. i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.345 − 0.598i)3-s + (−0.249 + 0.433i)4-s + (−0.949 − 1.64i)5-s − 0.488·6-s + (0.959 + 0.280i)7-s + 0.353·8-s + (0.261 + 0.452i)9-s + (−0.671 + 1.16i)10-s + (0.116 − 0.201i)11-s + (0.172 + 0.299i)12-s + 1.08·13-s + (−0.167 − 0.686i)14-s − 1.31·15-s + (−0.125 − 0.216i)16-s + (−0.191 + 0.331i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-0.340 + 0.940i$
Analytic conductor: \(2.24537\)
Root analytic conductor: \(1.49845\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :5/2),\ -0.340 + 0.940i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.635934 - 0.906946i\)
\(L(\frac12)\) \(\approx\) \(0.635934 - 0.906946i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 3.46i)T \)
7 \( 1 + (-124. - 36.3i)T \)
good3 \( 1 + (-5.38 + 9.33i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (53.0 + 91.8i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-46.7 + 80.9i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 661.T + 3.71e5T^{2} \)
17 \( 1 + (227. - 394. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (553. + 958. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (374. + 648. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 2.80e3T + 2.05e7T^{2} \)
31 \( 1 + (-179. + 311. i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-3.40e3 - 5.90e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 2.31e3T + 1.15e8T^{2} \)
43 \( 1 + 1.99e4T + 1.47e8T^{2} \)
47 \( 1 + (-7.10e3 - 1.23e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.30e4 - 2.26e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (2.45e3 - 4.24e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-6.60e3 - 1.14e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.98e4 + 5.16e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 8.90e3T + 1.80e9T^{2} \)
73 \( 1 + (-5.24e3 + 9.07e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-3.61e3 - 6.26e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.00e5T + 3.93e9T^{2} \)
89 \( 1 + (1.00e4 + 1.73e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 2.33e4T + 8.58e9T^{2} \)
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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

−18.45149740555795347775875119155, −16.96050749275179679923601118275, −15.67829701379005725946976869045, −13.52239750382337875824058026308, −12.44049698103609395851768713103, −11.21325209236766362741880039523, −8.709011740686337216029476038173, −8.038429067512796837278097485498, −4.56279838356865877569943019482, −1.30365111572807603828248578951, 3.88711556455516697355700158306, 6.79317747090843123628492547931, 8.235426131881934918532940546353, 10.25775630975430484468723848050, 11.41626777088793268351708893784, 14.20892198817361514234085594807, 15.00480933189398049561018529272, 15.91286631367618832483010908702, 17.88536383172875995959732915550, 18.67273827793102064621438207219

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