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Properties

Label 2-103-103.46-c3-0-2
Degree $2$
Conductor $103$
Sign $0.0799 + 0.996i$
Analytic cond. $6.07719$
Root an. cond. $2.46519$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.46 + 4.26i)2-s − 9.26·3-s + (−8.10 + 14.0i)4-s + (−6.00 + 10.4i)5-s + (−22.8 − 39.4i)6-s + (15.4 − 26.7i)7-s − 40.4·8-s + 58.8·9-s − 59.1·10-s + (−15.0 + 26.0i)11-s + (75.1 − 130. i)12-s − 51.5·13-s + 152.·14-s + (55.6 − 96.4i)15-s + (−34.6 − 60.0i)16-s + (−7.56 + 13.1i)17-s + ⋯
L(s)  = 1  + (0.869 + 1.50i)2-s − 1.78·3-s + (−1.01 + 1.75i)4-s + (−0.537 + 0.930i)5-s + (−1.55 − 2.68i)6-s + (0.833 − 1.44i)7-s − 1.78·8-s + 2.18·9-s − 1.86·10-s + (−0.412 + 0.714i)11-s + (1.80 − 3.13i)12-s − 1.09·13-s + 2.90·14-s + (0.958 − 1.65i)15-s + (−0.541 − 0.938i)16-s + (−0.107 + 0.187i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0799 + 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0799 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(103\)
Sign: $0.0799 + 0.996i$
Analytic conductor: \(6.07719\)
Root analytic conductor: \(2.46519\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 103,\ (\ :3/2),\ 0.0799 + 0.996i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.257333 - 0.237516i\)
\(L(\frac12)\) \(\approx\) \(0.257333 - 0.237516i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 + (-379. - 973. i)T \)
good2 \( 1 + (-2.46 - 4.26i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + 9.26T + 27T^{2} \)
5 \( 1 + (6.00 - 10.4i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-15.4 + 26.7i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (15.0 - 26.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 51.5T + 2.19e3T^{2} \)
17 \( 1 + (7.56 - 13.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (69.3 + 120. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 153.T + 1.21e4T^{2} \)
29 \( 1 + (-10.2 - 17.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 57.8T + 2.97e4T^{2} \)
37 \( 1 + 132.T + 5.06e4T^{2} \)
41 \( 1 + (-59.0 - 102. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-139. - 241. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (27.8 - 48.2i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (263. - 456. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-258. - 448. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + 494.T + 2.26e5T^{2} \)
67 \( 1 + (-299. + 519. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (120. - 208. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 674.T + 3.89e5T^{2} \)
79 \( 1 - 521.T + 4.93e5T^{2} \)
83 \( 1 + (581. + 1.00e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 23.0T + 7.04e5T^{2} \)
97 \( 1 + (356. + 616. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.36367255377047804264357494042, −13.20156058389522067808141028276, −12.16010518790477868038235442412, −11.05978365218155418569900966868, −10.29718555562710582501216379473, −7.61266502801168639033448866383, −7.20204962863076848685296077429, −6.30415858293890244359179935436, −4.80614646684215831839157923599, −4.34620561569618577225243757310, 0.18200568537791100313882054896, 1.91640825358006317271747691324, 4.30232456108700925712687296834, 5.20478909567539206826036854754, 5.83134153204581178912610652705, 8.278860593024999451376963495291, 9.903786113978799397310962505320, 10.91056713075727565543602075975, 11.90012637534983406751038158418, 12.13393859162555994081995405116

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