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Properties

Label 2-115-5.4-c1-0-9
Degree $2$
Conductor $115$
Sign $-0.976 + 0.214i$
Analytic cond. $0.918279$
Root an. cond. $0.958269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.37i·2-s − 1.95i·3-s − 3.66·4-s + (0.479 + 2.18i)5-s − 4.66·6-s − 2.28i·7-s + 3.95i·8-s − 0.840·9-s + (5.19 − 1.14i)10-s + 1.12·11-s + 7.17i·12-s + 5.95i·13-s − 5.43·14-s + (4.27 − 0.940i)15-s + 2.09·16-s − 5.80i·17-s + ⋯
L(s)  = 1  − 1.68i·2-s − 1.13i·3-s − 1.83·4-s + (0.214 + 0.976i)5-s − 1.90·6-s − 0.863i·7-s + 1.39i·8-s − 0.280·9-s + (1.64 − 0.361i)10-s + 0.338·11-s + 2.07i·12-s + 1.65i·13-s − 1.45·14-s + (1.10 − 0.242i)15-s + 0.523·16-s − 1.40i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.976 + 0.214i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1/2),\ -0.976 + 0.214i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.108893 - 1.00305i\)
\(L(\frac12)\) \(\approx\) \(0.108893 - 1.00305i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.479 - 2.18i)T \)
23 \( 1 + iT \)
good2 \( 1 + 2.37iT - 2T^{2} \)
3 \( 1 + 1.95iT - 3T^{2} \)
7 \( 1 + 2.28iT - 7T^{2} \)
11 \( 1 - 1.12T + 11T^{2} \)
13 \( 1 - 5.95iT - 13T^{2} \)
17 \( 1 + 5.80iT - 17T^{2} \)
19 \( 1 - 4.08T + 19T^{2} \)
29 \( 1 - 0.408T + 29T^{2} \)
31 \( 1 + 3.19T + 31T^{2} \)
37 \( 1 - 9.80iT - 37T^{2} \)
41 \( 1 - 6.27T + 41T^{2} \)
43 \( 1 - 7.75iT - 43T^{2} \)
47 \( 1 - 6.40iT - 47T^{2} \)
53 \( 1 + 6.73iT - 53T^{2} \)
59 \( 1 - 4.75T + 59T^{2} \)
61 \( 1 + 6.33T + 61T^{2} \)
67 \( 1 + 0.283iT - 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 - 9.61iT - 73T^{2} \)
79 \( 1 + 4.48T + 79T^{2} \)
83 \( 1 + 10.8iT - 83T^{2} \)
89 \( 1 + 5.68T + 89T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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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

−13.01594372965946494867125097818, −11.68780640141083658914923861794, −11.44687200453004858254690928588, −10.11003503590055307712284487832, −9.299929702622690237451935480432, −7.42847407166445297181581665105, −6.64728380596904064174942250890, −4.32929558696145820560478787902, −2.83555660150489837730819040837, −1.42939676877140458490427123424, 3.99893594056389816868909038101, 5.38005674754705759608137432761, 5.74132543623244504221758195139, 7.63989835367976680665340664785, 8.700506251679590464744058463536, 9.336111704148776990172713578344, 10.51656850836679121150672956589, 12.34488318098791564667308873126, 13.25664624055274704126806561614, 14.58474084123959755208240556738

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