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Properties

Label 2-1755-65.38-c0-0-1
Degree $2$
Conductor $1755$
Sign $0.632 - 0.774i$
Analytic cond. $0.875859$
Root an. cond. $0.935873$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.184 + 0.184i)2-s − 0.931i·4-s + (−0.130 + 0.991i)5-s + (0.356 − 0.356i)8-s + (−0.207 + 0.158i)10-s + 1.84i·11-s + (0.707 + 0.707i)13-s − 0.800·16-s + (0.923 + 0.121i)20-s + (−0.341 + 0.341i)22-s + (−0.965 − 0.258i)25-s + 0.261i·26-s + (−0.504 − 0.504i)32-s + (0.307 + 0.400i)40-s − 0.765i·41-s + ⋯
L(s)  = 1  + (0.184 + 0.184i)2-s − 0.931i·4-s + (−0.130 + 0.991i)5-s + (0.356 − 0.356i)8-s + (−0.207 + 0.158i)10-s + 1.84i·11-s + (0.707 + 0.707i)13-s − 0.800·16-s + (0.923 + 0.121i)20-s + (−0.341 + 0.341i)22-s + (−0.965 − 0.258i)25-s + 0.261i·26-s + (−0.504 − 0.504i)32-s + (0.307 + 0.400i)40-s − 0.765i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $0.632 - 0.774i$
Analytic conductor: \(0.875859\)
Root analytic conductor: \(0.935873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1755} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1755,\ (\ :0),\ 0.632 - 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.217979538\)
\(L(\frac12)\) \(\approx\) \(1.217979538\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.130 - 0.991i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-0.184 - 0.184i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 - 1.84iT - T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + 0.765iT - T^{2} \)
43 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
47 \( 1 + (-1.40 - 1.40i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - 1.21T + T^{2} \)
61 \( 1 + 0.517T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.98iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (0.860 - 0.860i)T - iT^{2} \)
89 \( 1 + 1.58T + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.648762455040395260864101061220, −9.089743274187394467141596071793, −7.72785801565488037002947802199, −7.08517738347877354236143628765, −6.47630152324525851999302308271, −5.70320575355055898737665403308, −4.58930253998890215787055693305, −3.99318502472157615800235433469, −2.53010478163028528217368639334, −1.62704196287770318835448944131, 0.949037432563019967903313655673, 2.59660159175689210524848333822, 3.57901033122961042473876256495, 4.16187457761220008953059570908, 5.39500942928424334355471638674, 5.91309608860761367583304533394, 7.17166812210607287676765172712, 8.068015307643718656595470553255, 8.595094856554314342094006979034, 8.977085723609595116227123538469

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