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Properties

Label 2-57-19.7-c1-0-0
Degree $2$
Conductor $57$
Sign $0.254 - 0.967i$
Analytic cond. $0.455147$
Root an. cond. $0.674646$
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  + (1.25 + 2.17i)2-s + (−0.5 − 0.866i)3-s + (−2.16 + 3.74i)4-s + (−1.66 − 2.87i)5-s + (1.25 − 2.17i)6-s + 2.32·7-s − 5.83·8-s + (−0.499 + 0.866i)9-s + (4.17 − 7.23i)10-s − 1.70·11-s + 4.32·12-s + (−2.01 + 3.48i)13-s + (2.91 + 5.05i)14-s + (−1.66 + 2.87i)15-s + (−3.01 − 5.22i)16-s + ⋯
L(s)  = 1  + (0.888 + 1.53i)2-s + (−0.288 − 0.499i)3-s + (−1.08 + 1.87i)4-s + (−0.742 − 1.28i)5-s + (0.513 − 0.888i)6-s + 0.877·7-s − 2.06·8-s + (−0.166 + 0.288i)9-s + (1.32 − 2.28i)10-s − 0.514·11-s + 1.24·12-s + (−0.558 + 0.967i)13-s + (0.779 + 1.35i)14-s + (−0.428 + 0.742i)15-s + (−0.753 − 1.30i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $0.254 - 0.967i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{57} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :1/2),\ 0.254 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.846487 + 0.652629i\)
\(L(\frac12)\) \(\approx\) \(0.846487 + 0.652629i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.193 + 4.35i)T \)
good2 \( 1 + (-1.25 - 2.17i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.66 + 2.87i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 2.32T + 7T^{2} \)
11 \( 1 + 1.70T + 11T^{2} \)
13 \( 1 + (2.01 - 3.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.17 + 2.03i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.32 - 5.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.70T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-3.32 - 5.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.353 + 0.612i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.98 + 8.62i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.853 + 1.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.69 - 2.93i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.18 + 7.25i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.70 - 8.15i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.82 + 10.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.67 - 2.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 + (-1.33 + 2.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.86 + 15.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.51868418320865219810418344004, −14.46507060263240882713199873838, −13.35633797664994695896872278126, −12.49214142205402292838886570007, −11.51545091123422372035104233252, −8.818723326978585894456721069922, −7.957880378968913562745530172828, −6.90563790575456069500084841087, −5.17781613996977954792779011068, −4.51778149694231630428310060811, 2.81871623103593173270887540885, 4.14833953162062799113937923518, 5.57222427665717340383168612318, 7.77450293761169913236384536454, 9.981722622473345269012849564950, 10.74316328484034785040043227968, 11.47038008537841220191263069048, 12.39486437786918420430776271376, 13.87997974762801974695047236037, 14.81197389192486419983691683028

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