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Properties

Label 2-930-15.8-c1-0-11
Degree $2$
Conductor $930$
Sign $-0.535 - 0.844i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  + (0.707 + 0.707i)2-s + (−0.750 − 1.56i)3-s + 1.00i·4-s + (−0.256 + 2.22i)5-s + (0.573 − 1.63i)6-s + (0.463 − 0.463i)7-s + (−0.707 + 0.707i)8-s + (−1.87 + 2.34i)9-s + (−1.75 + 1.38i)10-s + 0.686i·11-s + (1.56 − 0.750i)12-s + (−0.948 − 0.948i)13-s + 0.655·14-s + (3.66 − 1.26i)15-s − 1.00·16-s + (3.31 + 3.31i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.433 − 0.901i)3-s + 0.500i·4-s + (−0.114 + 0.993i)5-s + (0.233 − 0.667i)6-s + (0.175 − 0.175i)7-s + (−0.250 + 0.250i)8-s + (−0.624 + 0.781i)9-s + (−0.554 + 0.439i)10-s + 0.206i·11-s + (0.450 − 0.216i)12-s + (−0.263 − 0.263i)13-s + 0.175·14-s + (0.945 − 0.327i)15-s − 0.250·16-s + (0.803 + 0.803i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.535 - 0.844i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.535 - 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.580308 + 1.05495i\)
\(L(\frac12)\) \(\approx\) \(0.580308 + 1.05495i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (0.750 + 1.56i)T \)
5 \( 1 + (0.256 - 2.22i)T \)
31 \( 1 + T \)
good7 \( 1 + (-0.463 + 0.463i)T - 7iT^{2} \)
11 \( 1 - 0.686iT - 11T^{2} \)
13 \( 1 + (0.948 + 0.948i)T + 13iT^{2} \)
17 \( 1 + (-3.31 - 3.31i)T + 17iT^{2} \)
19 \( 1 + 0.717iT - 19T^{2} \)
23 \( 1 + (6.73 - 6.73i)T - 23iT^{2} \)
29 \( 1 + 5.30T + 29T^{2} \)
37 \( 1 + (5.56 - 5.56i)T - 37iT^{2} \)
41 \( 1 - 2.24iT - 41T^{2} \)
43 \( 1 + (-4.06 - 4.06i)T + 43iT^{2} \)
47 \( 1 + (-8.61 - 8.61i)T + 47iT^{2} \)
53 \( 1 + (-5.32 + 5.32i)T - 53iT^{2} \)
59 \( 1 - 13.6T + 59T^{2} \)
61 \( 1 + 13.2T + 61T^{2} \)
67 \( 1 + (-3.26 + 3.26i)T - 67iT^{2} \)
71 \( 1 + 6.49iT - 71T^{2} \)
73 \( 1 + (-1.62 - 1.62i)T + 73iT^{2} \)
79 \( 1 + 5.04iT - 79T^{2} \)
83 \( 1 + (6.41 - 6.41i)T - 83iT^{2} \)
89 \( 1 + 4.84T + 89T^{2} \)
97 \( 1 + (11.0 - 11.0i)T - 97iT^{2} \)
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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

−10.54519809695925443803791694517, −9.594244385407437830356987864622, −8.125513674367011145104114325589, −7.64854597336629833916145141949, −6.98508467825784367267641760027, −6.01361400314988944754401670737, −5.51198014180522343505180650514, −4.10241041678122279898783921778, −3.03934853752729025678650124874, −1.77396565737827392329760932814, 0.49353273521680293543123472234, 2.23464855256919300982907239435, 3.72771928135601505268707130536, 4.33110146767951576544137768359, 5.40051358682836061387490516021, 5.72084273043517862447418713081, 7.15216655820220638746444779201, 8.451416260288844917936745103675, 9.088584158124910411840866310449, 9.939001819766384193321892278289

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