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Properties

Label 2-14-7.2-c3-0-0
Degree $2$
Conductor $14$
Sign $0.605 - 0.795i$
Analytic cond. $0.826026$
Root an. cond. $0.908860$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (2.5 + 4.33i)3-s + (−1.99 − 3.46i)4-s + (4.5 − 7.79i)5-s − 10·6-s + (−14 − 12.1i)7-s + 7.99·8-s + (0.999 − 1.73i)9-s + (9 + 15.5i)10-s + (28.5 + 49.3i)11-s + (10 − 17.3i)12-s − 70·13-s + (35 − 12.1i)14-s + 45.0·15-s + (−8 + 13.8i)16-s + (−25.5 − 44.1i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.481 + 0.833i)3-s + (−0.249 − 0.433i)4-s + (0.402 − 0.697i)5-s − 0.680·6-s + (−0.755 − 0.654i)7-s + 0.353·8-s + (0.0370 − 0.0641i)9-s + (0.284 + 0.492i)10-s + (0.781 + 1.35i)11-s + (0.240 − 0.416i)12-s − 1.49·13-s + (0.668 − 0.231i)14-s + 0.774·15-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(0.826026\)
Root analytic conductor: \(0.908860\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :3/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.822479 + 0.407725i\)
\(L(\frac12)\) \(\approx\) \(0.822479 + 0.407725i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
7 \( 1 + (14 + 12.1i)T \)
good3 \( 1 + (-2.5 - 4.33i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-4.5 + 7.79i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-28.5 - 49.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 70T + 2.19e3T^{2} \)
17 \( 1 + (25.5 + 44.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (34.5 - 59.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 114T + 2.43e4T^{2} \)
31 \( 1 + (11.5 + 19.9i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-126.5 + 219. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 42T + 6.89e4T^{2} \)
43 \( 1 + 124T + 7.95e4T^{2} \)
47 \( 1 + (100.5 - 174. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-196.5 - 340. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (109.5 + 189. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-354.5 + 614. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (209.5 + 362. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 96T + 3.57e5T^{2} \)
73 \( 1 + (-156.5 - 271. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (230.5 - 399. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 588T + 5.71e5T^{2} \)
89 \( 1 + (-508.5 + 880. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.83e3T + 9.12e5T^{2} \)
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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

−19.70060149722936634821691663140, −17.62199509974744388632710877392, −16.69027607437785497004687294467, −15.43417976203765044553228414374, −14.26816558156304400322235008345, −12.56656183963219221409419477896, −9.850467289346286709120806720003, −9.368781952174122019062954225025, −7.08174257950588709211683065132, −4.56837402176893666395382714770, 2.65328915382901394963039405638, 6.62525929767708568596055772994, 8.545662963510313431303726219308, 10.17866315908668252753946338987, 11.98950982979369141487438511313, 13.29749318027771018416356786078, 14.54147840467397320909262650282, 16.60088533217385288874447020367, 18.14594227112232935887102024855, 19.18029280225304998422878326827

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