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Properties

Label 2-14-7.2-c5-0-1
Degree $2$
Conductor $14$
Sign $0.989 - 0.143i$
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 + (14.9 + 25.8i)3-s + (−7.99 − 13.8i)4-s + (10.5 − 18.1i)5-s + 119.·6-s + (−10.4 − 129. i)7-s − 63.9·8-s + (−322. + 558. i)9-s + (−42 − 72.7i)10-s + (−165. − 287. i)11-s + (238. − 412. i)12-s + 66.8·13-s + (−468. − 222. i)14-s + 625.·15-s + (−128 + 221. i)16-s + (−120. − 208. i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.955 + 1.65i)3-s + (−0.249 − 0.433i)4-s + (0.187 − 0.325i)5-s + 1.35·6-s + (−0.0803 − 0.996i)7-s − 0.353·8-s + (−1.32 + 2.29i)9-s + (−0.132 − 0.230i)10-s + (−0.413 − 0.716i)11-s + (0.477 − 0.827i)12-s + 0.109·13-s + (−0.638 − 0.303i)14-s + 0.718·15-s + (−0.125 + 0.216i)16-s + (−0.100 − 0.174i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $0.989 - 0.143i$
Analytic conductor: \(2.24537\)
Root analytic conductor: \(1.49845\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :5/2),\ 0.989 - 0.143i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.81296 + 0.130544i\)
\(L(\frac12)\) \(\approx\) \(1.81296 + 0.130544i\)
\(L(\frac{7}{2})\) 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 \( 1 + (10.4 + 129. i)T \)
good3 \( 1 + (-14.9 - 25.8i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-10.5 + 18.1i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (165. + 287. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 66.8T + 3.71e5T^{2} \)
17 \( 1 + (120. + 208. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (220. - 382. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-535. + 927. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 1.79e3T + 2.05e7T^{2} \)
31 \( 1 + (-2.84e3 - 4.92e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (5.60e3 - 9.70e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.20e4T + 1.15e8T^{2} \)
43 \( 1 + 9.92e3T + 1.47e8T^{2} \)
47 \( 1 + (-8.43e3 + 1.46e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (2.64e3 + 4.58e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (2.06e4 + 3.58e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.07e4 + 1.86e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.33e4 - 2.30e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 5.80e4T + 1.80e9T^{2} \)
73 \( 1 + (-1.99e4 - 3.46e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-2.19e4 + 3.80e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 2.24e4T + 3.93e9T^{2} \)
89 \( 1 + (1.20e4 - 2.08e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 7.18e4T + 8.58e9T^{2} \)
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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

−19.22694135258891909701693528861, −16.88367970922826953786440976195, −15.76379568474283236017430510140, −14.35353247030532887747563872279, −13.42212933569707119548334229621, −10.93617919733204394674060838961, −9.983951643573203375477369248476, −8.559133366651212841435419927124, −4.84809742668365262336310901976, −3.33101504595240378810891040191, 2.51779658033604245689268018810, 6.22842519636896385324830299461, 7.61079083741375787374510149104, 8.963875289832457624649367349786, 12.14620085077557419061462290641, 13.10071771384482645734271213575, 14.33343898144087727219906519046, 15.36412048777110957358309552441, 17.66920126648807823487630291292, 18.42802034860173992316376362552

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