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Properties

Label 2-230-5.4-c3-0-3
Degree $2$
Conductor $230$
Sign $-0.891 + 0.453i$
Analytic cond. $13.5704$
Root an. cond. $3.68380$
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  − 2i·2-s + 7.41i·3-s − 4·4-s + (−9.96 + 5.06i)5-s + 14.8·6-s + 26.6i·7-s + 8i·8-s − 27.9·9-s + (10.1 + 19.9i)10-s − 25.3·11-s − 29.6i·12-s − 60.6i·13-s + 53.2·14-s + (−37.5 − 73.8i)15-s + 16·16-s − 36.6i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.42i·3-s − 0.5·4-s + (−0.891 + 0.453i)5-s + 1.00·6-s + 1.43i·7-s + 0.353i·8-s − 1.03·9-s + (0.320 + 0.630i)10-s − 0.695·11-s − 0.713i·12-s − 1.29i·13-s + 1.01·14-s + (−0.646 − 1.27i)15-s + 0.250·16-s − 0.522i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.891 + 0.453i$
Analytic conductor: \(13.5704\)
Root analytic conductor: \(3.68380\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :3/2),\ -0.891 + 0.453i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0748929 - 0.312539i\)
\(L(\frac12)\) \(\approx\) \(0.0748929 - 0.312539i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
5 \( 1 + (9.96 - 5.06i)T \)
23 \( 1 - 23iT \)
good3 \( 1 - 7.41iT - 27T^{2} \)
7 \( 1 - 26.6iT - 343T^{2} \)
11 \( 1 + 25.3T + 1.33e3T^{2} \)
13 \( 1 + 60.6iT - 2.19e3T^{2} \)
17 \( 1 + 36.6iT - 4.91e3T^{2} \)
19 \( 1 - 49.1T + 6.85e3T^{2} \)
29 \( 1 + 6.51T + 2.43e4T^{2} \)
31 \( 1 + 145.T + 2.97e4T^{2} \)
37 \( 1 + 264. iT - 5.06e4T^{2} \)
41 \( 1 + 440.T + 6.89e4T^{2} \)
43 \( 1 - 50.0iT - 7.95e4T^{2} \)
47 \( 1 - 542. iT - 1.03e5T^{2} \)
53 \( 1 + 186. iT - 1.48e5T^{2} \)
59 \( 1 + 458.T + 2.05e5T^{2} \)
61 \( 1 - 104.T + 2.26e5T^{2} \)
67 \( 1 - 634. iT - 3.00e5T^{2} \)
71 \( 1 + 319.T + 3.57e5T^{2} \)
73 \( 1 + 731. iT - 3.89e5T^{2} \)
79 \( 1 + 6.71T + 4.93e5T^{2} \)
83 \( 1 - 950. iT - 5.71e5T^{2} \)
89 \( 1 - 690.T + 7.04e5T^{2} \)
97 \( 1 - 551. iT - 9.12e5T^{2} \)
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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

−12.05674600736077408112120314554, −11.18621010972679087238008921002, −10.49652463835127121040415338874, −9.576722745065480986493615736344, −8.699247204084594673174051212940, −7.67429314455549570341816081323, −5.62841307578357571175595651606, −4.88746942941806584108927297405, −3.49032271919283604832261577769, −2.75195657963237771987179689477, 0.13742200463570364744898224672, 1.44616729162520223199513554916, 3.72489807624011661745787094582, 4.90688966085201089294642715813, 6.53770012868787475513747200818, 7.24037269781465063380220507883, 7.83023902446074657627239281292, 8.757287437760907444033535597173, 10.24951645847900485395508890427, 11.48778008665620822935471520075

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