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Properties

Label 2-10e2-4.3-c4-0-8
Degree $2$
Conductor $100$
Sign $0.901 - 0.433i$
Analytic cond. $10.3369$
Root an. cond. $3.21511$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.90 + 0.888i)2-s − 12.9i·3-s + (14.4 − 6.93i)4-s + (11.5 + 50.6i)6-s + 78.0i·7-s + (−50.0 + 39.8i)8-s − 87.7·9-s + 95.6i·11-s + (−90.0 − 187. i)12-s + 159.·13-s + (−69.3 − 304. i)14-s + (159. − 199. i)16-s − 22.5·17-s + (342. − 77.9i)18-s + 324. i·19-s + ⋯
L(s)  = 1  + (−0.975 + 0.222i)2-s − 1.44i·3-s + (0.901 − 0.433i)4-s + (0.320 + 1.40i)6-s + 1.59i·7-s + (−0.782 + 0.622i)8-s − 1.08·9-s + 0.790i·11-s + (−0.625 − 1.30i)12-s + 0.944·13-s + (−0.353 − 1.55i)14-s + (0.624 − 0.780i)16-s − 0.0778·17-s + (1.05 − 0.240i)18-s + 0.900i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.901 - 0.433i$
Analytic conductor: \(10.3369\)
Root analytic conductor: \(3.21511\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :2),\ 0.901 - 0.433i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.985579 + 0.224541i\)
\(L(\frac12)\) \(\approx\) \(0.985579 + 0.224541i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.90 - 0.888i)T \)
5 \( 1 \)
good3 \( 1 + 12.9iT - 81T^{2} \)
7 \( 1 - 78.0iT - 2.40e3T^{2} \)
11 \( 1 - 95.6iT - 1.46e4T^{2} \)
13 \( 1 - 159.T + 2.85e4T^{2} \)
17 \( 1 + 22.5T + 8.35e4T^{2} \)
19 \( 1 - 324. iT - 1.30e5T^{2} \)
23 \( 1 - 204. iT - 2.79e5T^{2} \)
29 \( 1 - 295.T + 7.07e5T^{2} \)
31 \( 1 - 407. iT - 9.23e5T^{2} \)
37 \( 1 - 2.15e3T + 1.87e6T^{2} \)
41 \( 1 - 1.36e3T + 2.82e6T^{2} \)
43 \( 1 - 1.23e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.98e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.59e3T + 7.89e6T^{2} \)
59 \( 1 - 1.38e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.65e3T + 1.38e7T^{2} \)
67 \( 1 - 8.93e3iT - 2.01e7T^{2} \)
71 \( 1 + 3.37e3iT - 2.54e7T^{2} \)
73 \( 1 - 1.46e3T + 2.83e7T^{2} \)
79 \( 1 - 6.30e3iT - 3.89e7T^{2} \)
83 \( 1 - 6.10e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.70e3T + 6.27e7T^{2} \)
97 \( 1 - 1.98e3T + 8.85e7T^{2} \)
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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

−12.89893412782052417855416795392, −12.14751045232928089507001494122, −11.30825036605172262608543581475, −9.696931280725871197676554684852, −8.580386734316233855572643950781, −7.78922395661138296700219849162, −6.53767281893282287868304165504, −5.72540206392799341543379026041, −2.50262530100256592229275868168, −1.39379095357248223614729759491, 0.70281757939833512119028064828, 3.29863404150545850640459878063, 4.36244628798238487153048107887, 6.33582635270476942781957447426, 7.78324225559921132309720285285, 8.982200185568572061332749333392, 9.909801240598943200656027673584, 10.91677928839265141937976527732, 11.13933985184143734174519818999, 13.10575289828078696097515373677

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