Properties

Label 2-345-23.9-c1-0-11
Degree $2$
Conductor $345$
Sign $0.980 + 0.198i$
Analytic cond. $2.75483$
Root an. cond. $1.65977$
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  + (2.36 − 0.694i)2-s + (0.654 + 0.755i)3-s + (3.42 − 2.20i)4-s + (0.142 + 0.989i)5-s + (2.07 + 1.33i)6-s + (−0.477 + 1.04i)7-s + (3.34 − 3.86i)8-s + (−0.142 + 0.989i)9-s + (1.02 + 2.24i)10-s + (−3.93 − 1.15i)11-s + (3.90 + 1.14i)12-s + (−1.02 − 2.24i)13-s + (−0.403 + 2.80i)14-s + (−0.654 + 0.755i)15-s + (1.84 − 4.04i)16-s + (−2.49 − 1.60i)17-s + ⋯
L(s)  = 1  + (1.67 − 0.490i)2-s + (0.378 + 0.436i)3-s + (1.71 − 1.10i)4-s + (0.0636 + 0.442i)5-s + (0.846 + 0.543i)6-s + (−0.180 + 0.395i)7-s + (1.18 − 1.36i)8-s + (−0.0474 + 0.329i)9-s + (0.323 + 0.708i)10-s + (−1.18 − 0.348i)11-s + (1.12 + 0.331i)12-s + (−0.284 − 0.623i)13-s + (−0.107 + 0.750i)14-s + (−0.169 + 0.195i)15-s + (0.461 − 1.01i)16-s + (−0.604 − 0.388i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(345\)    =    \(3 \cdot 5 \cdot 23\)
Sign: $0.980 + 0.198i$
Analytic conductor: \(2.75483\)
Root analytic conductor: \(1.65977\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{345} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 345,\ (\ :1/2),\ 0.980 + 0.198i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.37007 - 0.337339i\)
\(L(\frac12)\) \(\approx\) \(3.37007 - 0.337339i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.654 - 0.755i)T \)
5 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (-3.26 + 3.51i)T \)
good2 \( 1 + (-2.36 + 0.694i)T + (1.68 - 1.08i)T^{2} \)
7 \( 1 + (0.477 - 1.04i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (3.93 + 1.15i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (1.02 + 2.24i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (2.49 + 1.60i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-0.574 + 0.369i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (0.0577 + 0.0371i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-3.86 + 4.45i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (1.27 - 8.85i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (0.643 + 4.47i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (1.18 + 1.36i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 - 2.37T + 47T^{2} \)
53 \( 1 + (5.73 - 12.5i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (-4.61 - 10.1i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-2.85 + 3.29i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (-3.11 + 0.914i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (13.0 - 3.81i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-4.62 + 2.97i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (0.0687 + 0.150i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-1.27 + 8.84i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (8.26 + 9.54i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-2.41 - 16.8i)T + (-93.0 + 27.3i)T^{2} \)
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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

−11.60290015431477800929146114737, −10.72301715721400651919618743221, −10.13159918282584879780461214433, −8.736184721560086754005482850770, −7.44880123364121975268677583846, −6.21290343531689255433811359440, −5.29602449431401725053605684999, −4.42485507676901241181000744752, −3.01800897711844701248300539265, −2.55938388470984909332802053889, 2.21608047828763582901740790395, 3.48881590012106713293267552273, 4.63015767769621412357586332739, 5.46627077344054948966234969877, 6.67756976863379684011048516531, 7.35576352401031575530910567006, 8.372028821854185102103256501561, 9.700549632163960206002095460999, 10.99605444874367868214733686661, 11.97938024714918761490041296531

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