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Properties

Label 2-230-5.4-c5-0-9
Degree $2$
Conductor $230$
Sign $-0.971 + 0.235i$
Analytic cond. $36.8882$
Root an. cond. $6.07357$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s − 1.04i·3-s − 16·4-s + (−54.3 + 13.1i)5-s + 4.18·6-s + 135. i·7-s − 64i·8-s + 241.·9-s + (−52.7 − 217. i)10-s + 306.·11-s + 16.7i·12-s + 653. i·13-s − 542.·14-s + (13.7 + 56.7i)15-s + 256·16-s + 535. i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.0670i·3-s − 0.5·4-s + (−0.971 + 0.235i)5-s + 0.0474·6-s + 1.04i·7-s − 0.353i·8-s + 0.995·9-s + (−0.166 − 0.687i)10-s + 0.764·11-s + 0.0335i·12-s + 1.07i·13-s − 0.740·14-s + (0.0158 + 0.0651i)15-s + 0.250·16-s + 0.449i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.235i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.971 + 0.235i$
Analytic conductor: \(36.8882\)
Root analytic conductor: \(6.07357\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :5/2),\ -0.971 + 0.235i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.103067278\)
\(L(\frac12)\) \(\approx\) \(1.103067278\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
5 \( 1 + (54.3 - 13.1i)T \)
23 \( 1 - 529iT \)
good3 \( 1 + 1.04iT - 243T^{2} \)
7 \( 1 - 135. iT - 1.68e4T^{2} \)
11 \( 1 - 306.T + 1.61e5T^{2} \)
13 \( 1 - 653. iT - 3.71e5T^{2} \)
17 \( 1 - 535. iT - 1.41e6T^{2} \)
19 \( 1 - 1.78e3T + 2.47e6T^{2} \)
29 \( 1 + 7.75e3T + 2.05e7T^{2} \)
31 \( 1 + 6.68e3T + 2.86e7T^{2} \)
37 \( 1 - 340. iT - 6.93e7T^{2} \)
41 \( 1 + 4.45e3T + 1.15e8T^{2} \)
43 \( 1 - 1.01e4iT - 1.47e8T^{2} \)
47 \( 1 + 3.57e3iT - 2.29e8T^{2} \)
53 \( 1 - 2.92e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.78e4T + 7.14e8T^{2} \)
61 \( 1 + 2.62e4T + 8.44e8T^{2} \)
67 \( 1 + 3.89e4iT - 1.35e9T^{2} \)
71 \( 1 + 8.16e3T + 1.80e9T^{2} \)
73 \( 1 + 6.73e3iT - 2.07e9T^{2} \)
79 \( 1 + 4.33e4T + 3.07e9T^{2} \)
83 \( 1 - 2.54e3iT - 3.93e9T^{2} \)
89 \( 1 + 3.23e4T + 5.58e9T^{2} \)
97 \( 1 - 2.54e4iT - 8.58e9T^{2} \)
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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

−11.95311517673238647405603059959, −11.07325214910941453847215268317, −9.529304194456869647921197431992, −8.941418496471001329164730920783, −7.66427972791793057753536116892, −6.99307137810138059192201220059, −5.85265645838085102813682835231, −4.51540361258754231207862501524, −3.53319583375219527730484197391, −1.58946815502936797617403341735, 0.35939003072983228474930604164, 1.37330630596199239822492067009, 3.42751965867767946815958705670, 4.06208476649959147224849309314, 5.24093180002217541445686517988, 7.14809152794712692010181835817, 7.68332397941879824570662180312, 9.052231158679167156033573141305, 10.00058903072355060506588909267, 10.88859658138825944425879514218

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