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Properties

Label 2-14-7.3-c2-0-1
Degree $2$
Conductor $14$
Sign $0.900 + 0.435i$
Analytic cond. $0.381472$
Root an. cond. $0.617634$
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 + (0.621 + 0.358i)3-s + (−0.999 + 1.73i)4-s + (−5.74 + 3.31i)5-s − 1.01i·6-s + (6.24 − 3.16i)7-s + 2.82·8-s + (−4.24 − 7.34i)9-s + (8.12 + 4.68i)10-s + (2.37 − 4.11i)11-s + (−1.24 + 0.717i)12-s + 15.2i·13-s + (−8.29 − 5.40i)14-s − 4.75·15-s + (−2.00 − 3.46i)16-s + (−3.25 − 1.88i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.207 + 0.119i)3-s + (−0.249 + 0.433i)4-s + (−1.14 + 0.663i)5-s − 0.169i·6-s + (0.891 − 0.452i)7-s + 0.353·8-s + (−0.471 − 0.816i)9-s + (0.812 + 0.468i)10-s + (0.216 − 0.374i)11-s + (−0.103 + 0.0597i)12-s + 1.17i·13-s + (−0.592 − 0.386i)14-s − 0.317·15-s + (−0.125 − 0.216i)16-s + (−0.191 − 0.110i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $0.900 + 0.435i$
Analytic conductor: \(0.381472\)
Root analytic conductor: \(0.617634\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :1),\ 0.900 + 0.435i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.623332 - 0.142984i\)
\(L(\frac12)\) \(\approx\) \(0.623332 - 0.142984i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
7 \( 1 + (-6.24 + 3.16i)T \)
good3 \( 1 + (-0.621 - 0.358i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (5.74 - 3.31i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.37 + 4.11i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 15.2iT - 169T^{2} \)
17 \( 1 + (3.25 + 1.88i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-3.62 + 2.09i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-13.8 - 24.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 3.51T + 841T^{2} \)
31 \( 1 + (42.3 + 24.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-1.47 - 2.54i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 27.9iT - 1.68e3T^{2} \)
43 \( 1 + 10.4T + 1.84e3T^{2} \)
47 \( 1 + (-45.6 + 26.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (27.9 - 48.4i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-33.5 - 19.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (78.3 - 45.2i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-17.3 + 29.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 36.4T + 5.04e3T^{2} \)
73 \( 1 + (-45.5 - 26.3i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-16.8 - 29.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 127. iT - 6.88e3T^{2} \)
89 \( 1 + (43.5 - 25.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 101. iT - 9.40e3T^{2} \)
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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

−19.45989058223634010030560243418, −18.34685901321837799123077444948, −16.95261971282567295090440228291, −15.22279019022005097097799665439, −14.02141981904875035769810097803, −11.78489684476757193141817960336, −11.09874051596798499778823202451, −9.012872723719945435784824270220, −7.38191162710491748042669853723, −3.81523649303225126491768400041, 5.02126322441029586578365387396, 7.75089215970672930435832883388, 8.625141808881862966633483659949, 10.99442918404639935465779445083, 12.60275105804830871495238879448, 14.49967147108960395923192122546, 15.59843740534224309097094919273, 16.83979611004929080851080610240, 18.20007997720568305996118873839, 19.59326568482731538094486456532

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