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Properties

Label 2-714-119.72-c1-0-18
Degree $2$
Conductor $714$
Sign $0.948 - 0.316i$
Analytic cond. $5.70131$
Root an. cond. $2.38774$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.965 + 0.258i)3-s + (0.499 + 0.866i)4-s + (−0.258 − 0.965i)5-s + (0.707 + 0.707i)6-s + (2.43 − 1.02i)7-s + 0.999i·8-s + (0.866 + 0.499i)9-s + (0.258 − 0.965i)10-s + (0.669 − 2.49i)11-s + (0.258 + 0.965i)12-s − 1.82·13-s + (2.62 + 0.331i)14-s − i·15-s + (−0.5 + 0.866i)16-s + (4.12 + 0.0693i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.557 + 0.149i)3-s + (0.249 + 0.433i)4-s + (−0.115 − 0.431i)5-s + (0.288 + 0.288i)6-s + (0.921 − 0.387i)7-s + 0.353i·8-s + (0.288 + 0.166i)9-s + (0.0818 − 0.305i)10-s + (0.201 − 0.753i)11-s + (0.0747 + 0.278i)12-s − 0.507·13-s + (0.701 + 0.0887i)14-s − 0.258i·15-s + (−0.125 + 0.216i)16-s + (0.999 + 0.0168i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(714\)    =    \(2 \cdot 3 \cdot 7 \cdot 17\)
Sign: $0.948 - 0.316i$
Analytic conductor: \(5.70131\)
Root analytic conductor: \(2.38774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{714} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 714,\ (\ :1/2),\ 0.948 - 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.75480 + 0.447206i\)
\(L(\frac12)\) \(\approx\) \(2.75480 + 0.447206i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (-2.43 + 1.02i)T \)
17 \( 1 + (-4.12 - 0.0693i)T \)
good5 \( 1 + (0.258 + 0.965i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.669 + 2.49i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + 1.82T + 13T^{2} \)
19 \( 1 + (-3.46 - 2i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.42 - 1.18i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.87 - 1.87i)T - 29iT^{2} \)
31 \( 1 + (4.26 + 1.14i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.31 - 4.89i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (6.94 + 6.94i)T + 41iT^{2} \)
43 \( 1 - 12.2iT - 43T^{2} \)
47 \( 1 + (-0.378 + 0.655i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.97 - 2.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.55 + 3.20i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.49 + 0.669i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.58 + 2.74i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.41 - 6.41i)T - 71iT^{2} \)
73 \( 1 + (8.76 + 2.34i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.79 - 1.55i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + 9.58iT - 83T^{2} \)
89 \( 1 + (4.82 - 8.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.48 - 8.48i)T - 97iT^{2} \)
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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

−10.46390582558111232014221047589, −9.548409281922358626695202156920, −8.444429327988683751018263570406, −7.924375340210900005760459359535, −7.11940649386936925232139408652, −5.78142767897613144991518394791, −4.99285372053320597948938384849, −4.02697735879659951931484114419, −3.07870366935278180604352741720, −1.48373649611569649848319343858, 1.63109753450818900483209509434, 2.65996232669202958307926568611, 3.76124446054696414730355300756, 4.84768494796051252499877842549, 5.68415520869087393245409149553, 7.06177988724090170661690597949, 7.58590650413932927448495188183, 8.677970344795849241717970045574, 9.661150054686211403045963319113, 10.38286185080379390537080364058

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