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Properties

Label 2-930-155.129-c1-0-2
Degree $2$
Conductor $930$
Sign $-0.605 - 0.795i$
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 + (−0.866 − 0.5i)3-s − 4-s + (1.24 + 1.85i)5-s + (−0.5 + 0.866i)6-s + (−0.669 − 0.386i)7-s + i·8-s + (0.499 + 0.866i)9-s + (1.85 − 1.24i)10-s + (0.0912 + 0.157i)11-s + (0.866 + 0.5i)12-s + (−3.35 + 1.93i)13-s + (−0.386 + 0.669i)14-s + (−0.154 − 2.23i)15-s + 16-s + (−5.39 − 3.11i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.499 − 0.288i)3-s − 0.5·4-s + (0.558 + 0.829i)5-s + (−0.204 + 0.353i)6-s + (−0.253 − 0.146i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.586 − 0.394i)10-s + (0.0274 + 0.0476i)11-s + (0.249 + 0.144i)12-s + (−0.929 + 0.536i)13-s + (−0.103 + 0.178i)14-s + (−0.0398 − 0.575i)15-s + 0.250·16-s + (−1.30 − 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0528234 + 0.106563i\)
\(L(\frac12)\) \(\approx\) \(0.0528234 + 0.106563i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-1.24 - 1.85i)T \)
31 \( 1 + (4.63 + 3.08i)T \)
good7 \( 1 + (0.669 + 0.386i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0912 - 0.157i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.35 - 1.93i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.39 + 3.11i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.298 - 0.517i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.42iT - 23T^{2} \)
29 \( 1 + 7.12T + 29T^{2} \)
37 \( 1 + (5.54 + 3.20i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.930 - 1.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.78 - 3.34i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.17iT - 47T^{2} \)
53 \( 1 + (7.49 - 4.32i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.03 - 10.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 0.975T + 61T^{2} \)
67 \( 1 + (0.371 - 0.214i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.02 - 6.97i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-9.05 + 5.23i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.64 + 6.30i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.97 - 4.02i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.839T + 89T^{2} \)
97 \( 1 - 9.51iT - 97T^{2} \)
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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

−10.59248612671350842182622560835, −9.539048648864866269190374847170, −9.154026876344737104026976878070, −7.63182666221366837502601510981, −6.93699354813059958966277625317, −6.10975444471342734664110970482, −5.05905310497463377318892450153, −4.05189757266312140603943400680, −2.70813969277236995885329348143, −1.91003901626441110257890980420, 0.05535163578201873735554335384, 1.91556881725390346209082743022, 3.66017128078777135098319586833, 4.78717006749754049491659878942, 5.39549221527307043865516837590, 6.20326703759808271767123772843, 7.09947965946576209062818175036, 8.124458358284943013726908093909, 9.060786383292358835254723934376, 9.539017794326174048294881538258

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