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Properties

Label 2-357-119.67-c1-0-13
Degree $2$
Conductor $357$
Sign $0.824 - 0.566i$
Analytic cond. $2.85065$
Root an. cond. $1.68838$
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  + (0.909 + 1.57i)2-s + (−0.866 − 0.5i)3-s + (−0.652 + 1.13i)4-s + (1.17 − 0.678i)5-s − 1.81i·6-s + (0.327 − 2.62i)7-s + 1.26·8-s + (0.499 + 0.866i)9-s + (2.13 + 1.23i)10-s + (0.471 + 0.272i)11-s + (1.13 − 0.652i)12-s + 0.514·13-s + (4.43 − 1.87i)14-s − 1.35·15-s + (2.45 + 4.24i)16-s + (3.97 − 1.08i)17-s + ⋯
L(s)  = 1  + (0.642 + 1.11i)2-s + (−0.499 − 0.288i)3-s + (−0.326 + 0.565i)4-s + (0.525 − 0.303i)5-s − 0.742i·6-s + (0.123 − 0.992i)7-s + 0.446·8-s + (0.166 + 0.288i)9-s + (0.675 + 0.390i)10-s + (0.142 + 0.0821i)11-s + (0.326 − 0.188i)12-s + 0.142·13-s + (1.18 − 0.500i)14-s − 0.350·15-s + (0.613 + 1.06i)16-s + (0.964 − 0.262i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $0.824 - 0.566i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ 0.824 - 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78663 + 0.554426i\)
\(L(\frac12)\) \(\approx\) \(1.78663 + 0.554426i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.327 + 2.62i)T \)
17 \( 1 + (-3.97 + 1.08i)T \)
good2 \( 1 + (-0.909 - 1.57i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.17 + 0.678i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.471 - 0.272i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.514T + 13T^{2} \)
19 \( 1 + (0.727 + 1.25i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.188 - 0.108i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.73iT - 29T^{2} \)
31 \( 1 + (2.91 + 1.68i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.82 - 5.67i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.22iT - 41T^{2} \)
43 \( 1 - 8.49T + 43T^{2} \)
47 \( 1 + (5.04 + 8.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.65 + 6.32i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.17 + 3.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.44 + 0.834i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.71 - 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.55iT - 71T^{2} \)
73 \( 1 + (-2.95 - 1.70i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (11.9 - 6.91i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.42T + 83T^{2} \)
89 \( 1 + (-0.962 - 1.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.58iT - 97T^{2} \)
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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

−11.62621065819321217648265476525, −10.60188392856495036773730929519, −9.813413556987236335446235879830, −8.391340252951972930796752278040, −7.34319297900746702951491671882, −6.77437019644127767665469764396, −5.65902979814996482739825883695, −4.99440673788137336836308591533, −3.78808610687145783111565198419, −1.43794193313118017162837713800, 1.77085943399575929906290175059, 2.98105254266059228056097551555, 4.15185204507904627387279893852, 5.38903488007414025609844802489, 6.08161091046500588513477744248, 7.56544829885375664293163201482, 8.896610564856893594512196916154, 9.944545910553770002994723178287, 10.61664827062007935498420556294, 11.46048129319331866479385574163

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