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Properties

Label 2-2175-5.4-c1-0-25
Degree $2$
Conductor $2175$
Sign $0.447 + 0.894i$
Analytic cond. $17.3674$
Root an. cond. $4.16742$
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  − 1.98i·2-s − i·3-s − 1.93·4-s − 1.98·6-s + 2.16i·7-s − 0.119i·8-s − 9-s + 3.44·11-s + 1.93i·12-s + 3.74i·13-s + 4.29·14-s − 4.11·16-s + 4.33i·17-s + 1.98i·18-s − 3.90·19-s + ⋯
L(s)  = 1  − 1.40i·2-s − 0.577i·3-s − 0.969·4-s − 0.810·6-s + 0.818i·7-s − 0.0423i·8-s − 0.333·9-s + 1.03·11-s + 0.559i·12-s + 1.03i·13-s + 1.14·14-s − 1.02·16-s + 1.05i·17-s + 0.467i·18-s − 0.895·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2175\)    =    \(3 \cdot 5^{2} \cdot 29\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(17.3674\)
Root analytic conductor: \(4.16742\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2175} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2175,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.656333095\)
\(L(\frac12)\) \(\approx\) \(1.656333095\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
29 \( 1 + T \)
good2 \( 1 + 1.98iT - 2T^{2} \)
7 \( 1 - 2.16iT - 7T^{2} \)
11 \( 1 - 3.44T + 11T^{2} \)
13 \( 1 - 3.74iT - 13T^{2} \)
17 \( 1 - 4.33iT - 17T^{2} \)
19 \( 1 + 3.90T + 19T^{2} \)
23 \( 1 - 3.78iT - 23T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 + 7.70iT - 37T^{2} \)
41 \( 1 - 7.88T + 41T^{2} \)
43 \( 1 + 0.975iT - 43T^{2} \)
47 \( 1 - 12.1iT - 47T^{2} \)
53 \( 1 - 13.1iT - 53T^{2} \)
59 \( 1 + 1.47T + 59T^{2} \)
61 \( 1 + 4.24T + 61T^{2} \)
67 \( 1 + 6.74iT - 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 12.3iT - 73T^{2} \)
79 \( 1 - 5.02T + 79T^{2} \)
83 \( 1 + 10.4iT - 83T^{2} \)
89 \( 1 + 4.35T + 89T^{2} \)
97 \( 1 - 3.75iT - 97T^{2} \)
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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

−9.146301521490862210793744339437, −8.488261443182238501037329919047, −7.41288768195546252697209858504, −6.39556158962876571017533101314, −5.98372758835787152325711404898, −4.46744238298049686312617556217, −3.91275194773925957129301406632, −2.74989407356915575498917317117, −1.98157822098564671085196866245, −1.17918097434105750771515152287, 0.65235584216096799006255581836, 2.56440043298082356769056289853, 3.78511780981422966365630727873, 4.61466438879545812686495472801, 5.23188119011665723039236717466, 6.37028954278872944143861292222, 6.66231662728529281964858147757, 7.64141465500375073726852332220, 8.301578776947915879886720936961, 8.939209343383113640063837920574

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