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Properties

Label 2-115-5.4-c3-0-24
Degree $2$
Conductor $115$
Sign $0.987 + 0.157i$
Analytic cond. $6.78521$
Root an. cond. $2.60484$
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  + 5.04i·2-s + 0.0620i·3-s − 17.4·4-s + (1.76 − 11.0i)5-s − 0.312·6-s − 30.6i·7-s − 47.5i·8-s + 26.9·9-s + (55.6 + 8.89i)10-s − 57.3·11-s − 1.08i·12-s − 4.79i·13-s + 154.·14-s + (0.685 + 0.109i)15-s + 100.·16-s − 62.0i·17-s + ⋯
L(s)  = 1  + 1.78i·2-s + 0.0119i·3-s − 2.17·4-s + (0.157 − 0.987i)5-s − 0.0212·6-s − 1.65i·7-s − 2.10i·8-s + 0.999·9-s + (1.76 + 0.281i)10-s − 1.57·11-s − 0.0260i·12-s − 0.102i·13-s + 2.95·14-s + (0.0117 + 0.00188i)15-s + 1.56·16-s − 0.884i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.987 + 0.157i$
Analytic conductor: \(6.78521\)
Root analytic conductor: \(2.60484\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3/2),\ 0.987 + 0.157i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.01768 - 0.0808038i\)
\(L(\frac12)\) \(\approx\) \(1.01768 - 0.0808038i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.76 + 11.0i)T \)
23 \( 1 - 23iT \)
good2 \( 1 - 5.04iT - 8T^{2} \)
3 \( 1 - 0.0620iT - 27T^{2} \)
7 \( 1 + 30.6iT - 343T^{2} \)
11 \( 1 + 57.3T + 1.33e3T^{2} \)
13 \( 1 + 4.79iT - 2.19e3T^{2} \)
17 \( 1 + 62.0iT - 4.91e3T^{2} \)
19 \( 1 + 99.2T + 6.85e3T^{2} \)
29 \( 1 - 220.T + 2.43e4T^{2} \)
31 \( 1 - 115.T + 2.97e4T^{2} \)
37 \( 1 - 79.9iT - 5.06e4T^{2} \)
41 \( 1 - 317.T + 6.89e4T^{2} \)
43 \( 1 - 62.9iT - 7.95e4T^{2} \)
47 \( 1 + 87.4iT - 1.03e5T^{2} \)
53 \( 1 + 662. iT - 1.48e5T^{2} \)
59 \( 1 + 523.T + 2.05e5T^{2} \)
61 \( 1 + 631.T + 2.26e5T^{2} \)
67 \( 1 + 371. iT - 3.00e5T^{2} \)
71 \( 1 - 231.T + 3.57e5T^{2} \)
73 \( 1 - 902. iT - 3.89e5T^{2} \)
79 \( 1 - 986.T + 4.93e5T^{2} \)
83 \( 1 + 18.5iT - 5.71e5T^{2} \)
89 \( 1 - 91.3T + 7.04e5T^{2} \)
97 \( 1 + 690. iT - 9.12e5T^{2} \)
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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

−13.37840173993185615417993281938, −12.73398178991900455058767058131, −10.51836549325334190646975079570, −9.652667013280650520834408771013, −8.214583417810759406948558516633, −7.57445419717510774822591291662, −6.53517861930441416616650368998, −4.97531593083864036025053460763, −4.35746853138681742895523426341, −0.55205010691470212965540742667, 2.10105657976187190291931239661, 2.86884476386074614648756617304, 4.55809969717977792985497664247, 6.09144558703921908115930711374, 8.072554016922785375259085806634, 9.261126495329450534393410105376, 10.41437538381899237327311978852, 10.77982060490544520191758814688, 12.20661604619336304392772281695, 12.66625794431732603852780896878

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