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Properties

Label 2-115-5.4-c3-0-0
Degree $2$
Conductor $115$
Sign $-0.615 - 0.788i$
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  − 3.60i·2-s + 8.84i·3-s − 4.97·4-s + (−8.81 + 6.87i)5-s + 31.8·6-s − 13.0i·7-s − 10.8i·8-s − 51.2·9-s + (24.7 + 31.7i)10-s − 43.2·11-s − 44.0i·12-s + 82.1i·13-s − 47.1·14-s + (−60.8 − 77.9i)15-s − 79.0·16-s − 15.5i·17-s + ⋯
L(s)  = 1  − 1.27i·2-s + 1.70i·3-s − 0.622·4-s + (−0.788 + 0.615i)5-s + 2.16·6-s − 0.707i·7-s − 0.481i·8-s − 1.89·9-s + (0.783 + 1.00i)10-s − 1.18·11-s − 1.05i·12-s + 1.75i·13-s − 0.900·14-s + (−1.04 − 1.34i)15-s − 1.23·16-s − 0.222i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.615 - 0.788i$
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: \((2,\ 115,\ (\ :3/2),\ -0.615 - 0.788i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.226278 + 0.463626i\)
\(L(\frac12)\) \(\approx\) \(0.226278 + 0.463626i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (8.81 - 6.87i)T \)
23 \( 1 - 23iT \)
good2 \( 1 + 3.60iT - 8T^{2} \)
3 \( 1 - 8.84iT - 27T^{2} \)
7 \( 1 + 13.0iT - 343T^{2} \)
11 \( 1 + 43.2T + 1.33e3T^{2} \)
13 \( 1 - 82.1iT - 2.19e3T^{2} \)
17 \( 1 + 15.5iT - 4.91e3T^{2} \)
19 \( 1 + 104.T + 6.85e3T^{2} \)
29 \( 1 + 53.8T + 2.43e4T^{2} \)
31 \( 1 - 270.T + 2.97e4T^{2} \)
37 \( 1 - 294. iT - 5.06e4T^{2} \)
41 \( 1 + 167.T + 6.89e4T^{2} \)
43 \( 1 + 51.4iT - 7.95e4T^{2} \)
47 \( 1 - 278. iT - 1.03e5T^{2} \)
53 \( 1 + 314. iT - 1.48e5T^{2} \)
59 \( 1 - 418.T + 2.05e5T^{2} \)
61 \( 1 + 27.1T + 2.26e5T^{2} \)
67 \( 1 - 823. iT - 3.00e5T^{2} \)
71 \( 1 - 93.0T + 3.57e5T^{2} \)
73 \( 1 - 908. iT - 3.89e5T^{2} \)
79 \( 1 + 84.2T + 4.93e5T^{2} \)
83 \( 1 + 1.25e3iT - 5.71e5T^{2} \)
89 \( 1 + 839.T + 7.04e5T^{2} \)
97 \( 1 + 169. iT - 9.12e5T^{2} \)
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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

−13.39513851933916643407575855977, −11.82466386258365231039559570628, −11.18261513702287304529537573095, −10.41115319430979734651128279146, −9.840650752818223282483343119871, −8.504898303460240900766736810356, −6.79685873661016073993377281792, −4.56607955048152554061483911691, −3.95233904955689385882109370161, −2.70930636188267423332404523031, 0.25875305451326157783871786753, 2.49861590458040917925219749707, 5.22849242860932700732858999977, 6.08408852615938380105704717272, 7.33546635132077155653059841758, 8.148628309348341350702329749256, 8.482465708601078179156144133730, 10.85929002269089767878490772150, 12.19592708704066024187486992545, 12.78766643550195931948892976837

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