Properties

Label 2-930-93.68-c1-0-37
Degree $2$
Conductor $930$
Sign $-0.743 + 0.668i$
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  i·2-s + (0.433 + 1.67i)3-s − 4-s + (0.866 + 0.5i)5-s + (1.67 − 0.433i)6-s + (−1.74 − 3.01i)7-s + i·8-s + (−2.62 + 1.45i)9-s + (0.5 − 0.866i)10-s + (2.28 − 3.95i)11-s + (−0.433 − 1.67i)12-s + (−6.16 − 3.55i)13-s + (−3.01 + 1.74i)14-s + (−0.463 + 1.66i)15-s + 16-s + (−1.01 − 1.75i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.250 + 0.968i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (0.684 − 0.176i)6-s + (−0.658 − 1.14i)7-s + 0.353i·8-s + (−0.874 + 0.484i)9-s + (0.158 − 0.273i)10-s + (0.688 − 1.19i)11-s + (−0.125 − 0.484i)12-s + (−1.70 − 0.986i)13-s + (−0.806 + 0.465i)14-s + (−0.119 + 0.430i)15-s + 0.250·16-s + (−0.245 − 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.743 + 0.668i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.743 + 0.668i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.292784 - 0.763659i\)
\(L(\frac12)\) \(\approx\) \(0.292784 - 0.763659i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.433 - 1.67i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (1.29 + 5.41i)T \)
good7 \( 1 + (1.74 + 3.01i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.28 + 3.95i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.16 + 3.55i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.01 + 1.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.87 - 6.71i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.42T + 23T^{2} \)
29 \( 1 + 6.35T + 29T^{2} \)
37 \( 1 + (-7.71 + 4.45i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.17 + 1.25i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.21 - 1.85i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.93iT - 47T^{2} \)
53 \( 1 + (-5.09 + 8.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.20 + 4.16i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 2.03iT - 61T^{2} \)
67 \( 1 + (3.08 - 5.35i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.26 + 1.30i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.81 - 4.51i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.33 - 4.23i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.54 + 7.87i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.74T + 89T^{2} \)
97 \( 1 - 5.16T + 97T^{2} \)
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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

−9.972254636494919797928342549785, −9.406166148568385428155320126844, −8.163099071054024756837516387842, −7.43295244294037907341540861601, −5.99578301924376073840797010437, −5.28123998802489163354517923443, −3.89524803902156678988670454682, −3.53753534994733640629685224003, −2.34859025341452319551903804474, −0.34878881029117349139276099012, 1.85668895558226997010534736608, 2.72145770959893369936817377787, 4.39283010965721001020249814395, 5.38082974863813860192736031125, 6.31713075691547501005889009712, 6.97922547937051480300619408596, 7.60642397761028455827681906880, 8.835984861483082191840495072477, 9.368400659176185934609726588098, 9.816150543483974049651770337248

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