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Properties

Label 2-105-7.2-c1-0-0
Degree $2$
Conductor $105$
Sign $-0.198 - 0.980i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
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 + 1.22i)2-s + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s − 1.41·6-s + (1.62 + 2.09i)7-s − 2.82·8-s + (−0.499 + 0.866i)9-s + (0.707 + 1.22i)10-s + (−1.70 − 2.95i)11-s − 1.58·13-s + (−3.70 + 0.507i)14-s + 0.999·15-s + (2.00 − 3.46i)16-s + (3.12 + 5.40i)17-s + (−0.707 − 1.22i)18-s + (3.32 − 5.76i)19-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)2-s + (0.288 + 0.499i)3-s + (0.223 − 0.387i)5-s − 0.577·6-s + (0.612 + 0.790i)7-s − 0.999·8-s + (−0.166 + 0.288i)9-s + (0.223 + 0.387i)10-s + (−0.514 − 0.891i)11-s − 0.439·13-s + (−0.990 + 0.135i)14-s + 0.258·15-s + (0.500 − 0.866i)16-s + (0.757 + 1.31i)17-s + (−0.166 − 0.288i)18-s + (0.763 − 1.32i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.198 - 0.980i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.589027 + 0.719951i\)
\(L(\frac12)\) \(\approx\) \(0.589027 + 0.719951i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-1.62 - 2.09i)T \)
good2 \( 1 + (0.707 - 1.22i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (1.70 + 2.95i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.58T + 13T^{2} \)
17 \( 1 + (-3.12 - 5.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.32 + 5.76i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.12 + 5.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.242T + 29T^{2} \)
31 \( 1 + (-0.0857 - 0.148i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.79 + 4.83i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.24T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + (4.65 - 8.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.585 + 1.01i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.707 + 1.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.24 - 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.86 + 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.41T + 71T^{2} \)
73 \( 1 + (1.03 + 1.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.32 + 4.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.41T + 83T^{2} \)
89 \( 1 + (-1.87 + 3.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−14.51544959982680661816495712870, −13.08445422300273091756294095398, −11.97049327166827792012884058517, −10.78455200582830370742280519049, −9.319754419749770283371809151912, −8.562070783046229850934412225092, −7.74998069104029779940496067301, −6.11473449179698966373873177186, −5.02312112477664554148778556602, −2.91151709495536626609309891081, 1.57664261147576100464138211758, 3.14266139598966056537513697754, 5.29006842559240291694108352725, 7.03228571844964354048694761145, 7.916902001221513787425387266516, 9.629025320752974606142441286694, 10.12979718346965109736494314099, 11.40393300258234978475603930212, 12.12566686586236568465496550320, 13.44492787300180295946487056082

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