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Properties

Label 2-930-465.464-c1-0-0
Degree $2$
Conductor $930$
Sign $-0.636 - 0.771i$
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  − 2-s + (0.230 − 1.71i)3-s + 4-s + (−0.286 + 2.21i)5-s + (−0.230 + 1.71i)6-s − 4.09i·7-s − 8-s + (−2.89 − 0.791i)9-s + (0.286 − 2.21i)10-s − 2.91·11-s + (0.230 − 1.71i)12-s − 2.84·13-s + 4.09i·14-s + (3.74 + 1.00i)15-s + 16-s + 3.40i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.133 − 0.991i)3-s + 0.5·4-s + (−0.128 + 0.991i)5-s + (−0.0941 + 0.700i)6-s − 1.54i·7-s − 0.353·8-s + (−0.964 − 0.263i)9-s + (0.0906 − 0.701i)10-s − 0.879·11-s + (0.0665 − 0.495i)12-s − 0.790·13-s + 1.09i·14-s + (0.965 + 0.258i)15-s + 0.250·16-s + 0.826i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.636 - 0.771i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.636 - 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00701466 + 0.0148722i\)
\(L(\frac12)\) \(\approx\) \(0.00701466 + 0.0148722i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.230 + 1.71i)T \)
5 \( 1 + (0.286 - 2.21i)T \)
31 \( 1 + (-4.53 - 3.23i)T \)
good7 \( 1 + 4.09iT - 7T^{2} \)
11 \( 1 + 2.91T + 11T^{2} \)
13 \( 1 + 2.84T + 13T^{2} \)
17 \( 1 - 3.40iT - 17T^{2} \)
19 \( 1 - 2.22T + 19T^{2} \)
23 \( 1 - 3.74iT - 23T^{2} \)
29 \( 1 + 5.67T + 29T^{2} \)
37 \( 1 - 1.51T + 37T^{2} \)
41 \( 1 + 0.438iT - 41T^{2} \)
43 \( 1 + 8.29T + 43T^{2} \)
47 \( 1 + 7.81T + 47T^{2} \)
53 \( 1 + 10.9iT - 53T^{2} \)
59 \( 1 + 2.26iT - 59T^{2} \)
61 \( 1 - 3.00iT - 61T^{2} \)
67 \( 1 - 9.68iT - 67T^{2} \)
71 \( 1 - 11.2iT - 71T^{2} \)
73 \( 1 + 2.62T + 73T^{2} \)
79 \( 1 - 15.1iT - 79T^{2} \)
83 \( 1 + 8.73iT - 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 - 3.32iT - 97T^{2} \)
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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

−10.21001900023664603198458768561, −9.842727871292396423590444882203, −8.336579155698940981687963036917, −7.71686056991535223693144848780, −7.14846006948375653520582272805, −6.58944353884984270371316581372, −5.38490548454853689612970517661, −3.74316475987063376473395100865, −2.80102950604139296747833375885, −1.55501837244211057650232396165, 0.009060799434287635031118297715, 2.21304173064258200133318446112, 3.07536554075301712879383166389, 4.71564898503173403236733886357, 5.23844684048603025068921823266, 6.09370872950998813168566795237, 7.68958733162239143349439943569, 8.295700065695611917723439288569, 9.148988265469135709283508577615, 9.490733309274943304716984859515

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