Properties

Label 2-294-7.4-c3-0-14
Degree $2$
Conductor $294$
Sign $0.605 + 0.795i$
Analytic cond. $17.3465$
Root an. cond. $4.16492$
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 + (−1.5 + 2.59i)3-s + (−1.99 + 3.46i)4-s + (−4 − 6.92i)5-s − 6·6-s − 7.99·8-s + (−4.5 − 7.79i)9-s + (7.99 − 13.8i)10-s + (−20 + 34.6i)11-s + (−6.00 − 10.3i)12-s + 4·13-s + 24·15-s + (−8 − 13.8i)16-s + (42 − 72.7i)17-s + (9 − 15.5i)18-s + (−74 − 128. i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.357 − 0.619i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.252 − 0.438i)10-s + (−0.548 + 0.949i)11-s + (−0.144 − 0.249i)12-s + 0.0853·13-s + 0.413·15-s + (−0.125 − 0.216i)16-s + (0.599 − 1.03i)17-s + (0.117 − 0.204i)18-s + (−0.893 − 1.54i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(17.3465\)
Root analytic conductor: \(4.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :3/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9480705900\)
\(L(\frac12)\) \(\approx\) \(0.9480705900\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 + (1.5 - 2.59i)T \)
7 \( 1 \)
good5 \( 1 + (4 + 6.92i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (20 - 34.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 4T + 2.19e3T^{2} \)
17 \( 1 + (-42 + 72.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (74 + 128. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (42 + 72.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 58T + 2.43e4T^{2} \)
31 \( 1 + (-68 + 117. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-111 - 192. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 420T + 6.89e4T^{2} \)
43 \( 1 + 164T + 7.95e4T^{2} \)
47 \( 1 + (244 + 422. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (239 - 413. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (274 - 474. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (346 + 599. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-454 + 786. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 524T + 3.57e5T^{2} \)
73 \( 1 + (220 - 381. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (608 + 1.05e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 684T + 5.71e5T^{2} \)
89 \( 1 + (302 + 523. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 832T + 9.12e5T^{2} \)
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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

−11.33668813506280532538537503202, −10.17327205337815577873249750054, −9.231594882267329263876464981298, −8.267948789633436406310289145424, −7.25643954537215780077776624808, −6.17062570241153316956728462472, −4.77684924651452475226130786052, −4.52140171966288868192363217261, −2.75456330137977340038610525385, −0.33752015792507152696621433121, 1.43973712575338838613327973047, 2.96603995273747560813670634733, 3.99815560761166531571864980821, 5.61003788151056195473730988689, 6.29313344313762492443785812238, 7.69848524372415640276775106786, 8.459118737224851503611659486033, 9.976374267976122901455145382724, 10.79034411428027791958373719113, 11.38543247386494693599503565393

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