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Properties

Label 2-930-15.2-c1-0-22
Degree $2$
Conductor $930$
Sign $-0.614 - 0.788i$
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 + (−0.919 + 1.46i)3-s − 1.00i·4-s + (2.22 + 0.261i)5-s + (−0.387 − 1.68i)6-s + (3.32 + 3.32i)7-s + (0.707 + 0.707i)8-s + (−1.30 − 2.69i)9-s + (−1.75 + 1.38i)10-s + 3.28i·11-s + (1.46 + 0.919i)12-s + (−1.75 + 1.75i)13-s − 4.70·14-s + (−2.42 + 3.01i)15-s − 1.00·16-s + (5.45 − 5.45i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.530 + 0.847i)3-s − 0.500i·4-s + (0.993 + 0.117i)5-s + (−0.158 − 0.689i)6-s + (1.25 + 1.25i)7-s + (0.250 + 0.250i)8-s + (−0.436 − 0.899i)9-s + (−0.555 + 0.438i)10-s + 0.989i·11-s + (0.423 + 0.265i)12-s + (−0.486 + 0.486i)13-s − 1.25·14-s + (−0.626 + 0.779i)15-s − 0.250·16-s + (1.32 − 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.614 - 0.788i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.614 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.611363 + 1.25159i\)
\(L(\frac12)\) \(\approx\) \(0.611363 + 1.25159i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.919 - 1.46i)T \)
5 \( 1 + (-2.22 - 0.261i)T \)
31 \( 1 - T \)
good7 \( 1 + (-3.32 - 3.32i)T + 7iT^{2} \)
11 \( 1 - 3.28iT - 11T^{2} \)
13 \( 1 + (1.75 - 1.75i)T - 13iT^{2} \)
17 \( 1 + (-5.45 + 5.45i)T - 17iT^{2} \)
19 \( 1 - 0.692iT - 19T^{2} \)
23 \( 1 + (-1.40 - 1.40i)T + 23iT^{2} \)
29 \( 1 - 1.76T + 29T^{2} \)
37 \( 1 + (-2.58 - 2.58i)T + 37iT^{2} \)
41 \( 1 + 6.83iT - 41T^{2} \)
43 \( 1 + (8.40 - 8.40i)T - 43iT^{2} \)
47 \( 1 + (-8.79 + 8.79i)T - 47iT^{2} \)
53 \( 1 + (4.69 + 4.69i)T + 53iT^{2} \)
59 \( 1 + 6.39T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + (7.69 + 7.69i)T + 67iT^{2} \)
71 \( 1 - 0.464iT - 71T^{2} \)
73 \( 1 + (-4.31 + 4.31i)T - 73iT^{2} \)
79 \( 1 - 14.1iT - 79T^{2} \)
83 \( 1 + (0.133 + 0.133i)T + 83iT^{2} \)
89 \( 1 + 5.90T + 89T^{2} \)
97 \( 1 + (2.86 + 2.86i)T + 97iT^{2} \)
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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

−10.07528004754262647464131845967, −9.510204380367418865977808688721, −8.992009030151211603104055102505, −7.917534463945231927360183377415, −6.90526322836651050274907543715, −5.86814151049833070441125170518, −5.13672469502096941177646519655, −4.75190752684440401200960717565, −2.78243444806922099229059774413, −1.59192372295170984088709376219, 0.950063375747558363245335359006, 1.61351951900543860168579389785, 3.00052650950125320246169762002, 4.50605896394153544154448177209, 5.50906926787933165122737751289, 6.33265646979482146429770980673, 7.50813299447732232311068466943, 7.985185141481514980177066468931, 8.808149149850193603196234589283, 10.17528000889550952622261786818

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