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Properties

Label 2-930-15.8-c1-0-18
Degree $2$
Conductor $930$
Sign $-0.229 + 0.973i$
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 + (−1.22 − 1.22i)3-s + 1.00i·4-s + (−2.12 + 0.707i)5-s + 1.73i·6-s + (−3.44 + 3.44i)7-s + (0.707 − 0.707i)8-s + 2.99i·9-s + (2 + 0.999i)10-s + 6.29i·11-s + (1.22 − 1.22i)12-s + (−4.44 − 4.44i)13-s + 4.87·14-s + (3.46 + 1.73i)15-s − 1.00·16-s + (−0.317 − 0.317i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.707 − 0.707i)3-s + 0.500i·4-s + (−0.948 + 0.316i)5-s + 0.707i·6-s + (−1.30 + 1.30i)7-s + (0.250 − 0.250i)8-s + 0.999i·9-s + (0.632 + 0.316i)10-s + 1.89i·11-s + (0.353 − 0.353i)12-s + (−1.23 − 1.23i)13-s + 1.30·14-s + (0.894 + 0.447i)15-s − 0.250·16-s + (−0.0770 − 0.0770i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.229 + 0.973i$
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: \((2,\ 930,\ (\ :1/2),\ -0.229 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147904 - 0.186885i\)
\(L(\frac12)\) \(\approx\) \(0.147904 - 0.186885i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 + (2.12 - 0.707i)T \)
31 \( 1 + T \)
good7 \( 1 + (3.44 - 3.44i)T - 7iT^{2} \)
11 \( 1 - 6.29iT - 11T^{2} \)
13 \( 1 + (4.44 + 4.44i)T + 13iT^{2} \)
17 \( 1 + (0.317 + 0.317i)T + 17iT^{2} \)
19 \( 1 + 4.89iT - 19T^{2} \)
23 \( 1 + (-1.73 + 1.73i)T - 23iT^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 2.82iT - 41T^{2} \)
43 \( 1 + (2.44 + 2.44i)T + 43iT^{2} \)
47 \( 1 + (-5.65 - 5.65i)T + 47iT^{2} \)
53 \( 1 + (4.24 - 4.24i)T - 53iT^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 - 0.449T + 61T^{2} \)
67 \( 1 + (3.55 - 3.55i)T - 67iT^{2} \)
71 \( 1 - 2.19iT - 71T^{2} \)
73 \( 1 + (-4.89 - 4.89i)T + 73iT^{2} \)
79 \( 1 - 0.898iT - 79T^{2} \)
83 \( 1 + (-6.29 + 6.29i)T - 83iT^{2} \)
89 \( 1 - 4.38T + 89T^{2} \)
97 \( 1 + (3 - 3i)T - 97iT^{2} \)
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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

−9.891213719440970046395614734951, −9.160610877116910192392999027739, −8.026236161603490950448765847899, −7.18320383467048436969815716724, −6.78818191730708150338787338657, −5.45838325145441194544805156229, −4.54889364543768827116125292604, −2.90776440489737960388774297876, −2.33764992848030806666331126506, −0.23092352624440512946637475216, 0.72886090260267993897011918759, 3.46624469834561626379446728126, 3.95228503883101307845995920765, 5.11102743372746100686758273185, 6.17606196886748479214224120273, 6.88084528445756814256661283140, 7.66711486777074360204181676572, 8.790993832505968208506798984242, 9.486572677784060266100040182938, 10.24426828167223196560919925799

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