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Properties

Label 2-230-5.4-c5-0-55
Degree $2$
Conductor $230$
Sign $0.508 - 0.861i$
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 − 27.5i·3-s − 16·4-s + (28.4 − 48.1i)5-s − 110.·6-s − 119. i·7-s + 64i·8-s − 514.·9-s + (−192. − 113. i)10-s − 12.4·11-s + 440. i·12-s + 426. i·13-s − 476.·14-s + (−1.32e3 − 782. i)15-s + 256·16-s + 1.32e3i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.76i·3-s − 0.5·4-s + (0.508 − 0.861i)5-s − 1.24·6-s − 0.919i·7-s + 0.353i·8-s − 2.11·9-s + (−0.608 − 0.359i)10-s − 0.0310·11-s + 0.883i·12-s + 0.699i·13-s − 0.650·14-s + (−1.52 − 0.898i)15-s + 0.250·16-s + 1.11i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.508 - 0.861i$
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: \((2,\ 230,\ (\ :5/2),\ 0.508 - 0.861i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9935766411\)
\(L(\frac12)\) \(\approx\) \(0.9935766411\)
\(L(\frac{7}{2})\) 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 \( 1 + (-28.4 + 48.1i)T \)
23 \( 1 + 529iT \)
good3 \( 1 + 27.5iT - 243T^{2} \)
7 \( 1 + 119. iT - 1.68e4T^{2} \)
11 \( 1 + 12.4T + 1.61e5T^{2} \)
13 \( 1 - 426. iT - 3.71e5T^{2} \)
17 \( 1 - 1.32e3iT - 1.41e6T^{2} \)
19 \( 1 - 674.T + 2.47e6T^{2} \)
29 \( 1 + 2.87e3T + 2.05e7T^{2} \)
31 \( 1 + 7.27e3T + 2.86e7T^{2} \)
37 \( 1 + 1.07e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.20e4T + 1.15e8T^{2} \)
43 \( 1 - 8.15e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.36e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.47e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.33e4T + 7.14e8T^{2} \)
61 \( 1 - 2.33e4T + 8.44e8T^{2} \)
67 \( 1 + 6.47e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.48e4T + 1.80e9T^{2} \)
73 \( 1 - 1.47e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.42e4T + 3.07e9T^{2} \)
83 \( 1 + 8.55e4iT - 3.93e9T^{2} \)
89 \( 1 + 8.94e4T + 5.58e9T^{2} \)
97 \( 1 + 2.76e4iT - 8.58e9T^{2} \)
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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

−10.77739523500870265440494820886, −9.471046238408614645288242663747, −8.481325670225592416822203228256, −7.57108326296979746667311054246, −6.51285799946494633962169827389, −5.38836379552558849976033940977, −3.83949735545247717501880886681, −2.06616604736391435101426095530, −1.37226131385584400785577579578, −0.28950630310988787880676609468, 2.70970963091712631046352104299, 3.70146774449462514295961608465, 5.27277982748045381941358757246, 5.57068952151461330859665930757, 7.03792303481939596554828546702, 8.470970181992524329941728538963, 9.402313655715638489663649896206, 9.959357293698610184396267637279, 10.92195694741905540588869198253, 11.84686732292156607110484654571

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