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Properties

Label 2-414-207.25-c1-0-23
Degree $2$
Conductor $414$
Sign $-0.690 + 0.723i$
Analytic cond. $3.30580$
Root an. cond. $1.81818$
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  + (0.888 − 0.458i)2-s + (0.143 − 1.72i)3-s + (0.580 − 0.814i)4-s + (−0.196 − 0.811i)5-s + (−0.663 − 1.60i)6-s + (−1.10 − 3.18i)7-s + (0.142 − 0.989i)8-s + (−2.95 − 0.496i)9-s + (−0.546 − 0.631i)10-s + (0.108 + 2.28i)11-s + (−1.32 − 1.11i)12-s + (−1.57 + 4.55i)13-s + (−2.44 − 2.32i)14-s + (−1.42 + 0.223i)15-s + (−0.327 − 0.945i)16-s + (2.81 − 6.15i)17-s + ⋯
L(s)  = 1  + (0.628 − 0.324i)2-s + (0.0829 − 0.996i)3-s + (0.290 − 0.407i)4-s + (−0.0880 − 0.362i)5-s + (−0.270 − 0.653i)6-s + (−0.417 − 1.20i)7-s + (0.0503 − 0.349i)8-s + (−0.986 − 0.165i)9-s + (−0.172 − 0.199i)10-s + (0.0328 + 0.689i)11-s + (−0.381 − 0.322i)12-s + (−0.437 + 1.26i)13-s + (−0.652 − 0.622i)14-s + (−0.368 + 0.0576i)15-s + (−0.0817 − 0.236i)16-s + (0.681 − 1.49i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.690 + 0.723i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ -0.690 + 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.694483 - 1.62343i\)
\(L(\frac12)\) \(\approx\) \(0.694483 - 1.62343i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.888 + 0.458i)T \)
3 \( 1 + (-0.143 + 1.72i)T \)
23 \( 1 + (-3.19 - 3.58i)T \)
good5 \( 1 + (0.196 + 0.811i)T + (-4.44 + 2.29i)T^{2} \)
7 \( 1 + (1.10 + 3.18i)T + (-5.50 + 4.32i)T^{2} \)
11 \( 1 + (-0.108 - 2.28i)T + (-10.9 + 1.04i)T^{2} \)
13 \( 1 + (1.57 - 4.55i)T + (-10.2 - 8.03i)T^{2} \)
17 \( 1 + (-2.81 + 6.15i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (-1.29 - 2.84i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (6.11 + 8.58i)T + (-9.48 + 27.4i)T^{2} \)
31 \( 1 + (-4.24 + 1.70i)T + (22.4 - 21.3i)T^{2} \)
37 \( 1 + (-10.1 + 2.96i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (1.55 + 6.41i)T + (-36.4 + 18.7i)T^{2} \)
43 \( 1 + (1.12 + 0.451i)T + (31.1 + 29.6i)T^{2} \)
47 \( 1 + (3.09 - 5.35i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.01 - 9.25i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (-2.75 + 7.94i)T + (-46.3 - 36.4i)T^{2} \)
61 \( 1 + (-9.72 - 7.64i)T + (14.3 + 59.2i)T^{2} \)
67 \( 1 + (0.245 - 5.16i)T + (-66.6 - 6.36i)T^{2} \)
71 \( 1 + (-3.46 - 2.22i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (0.979 + 2.14i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (-0.0200 + 0.00387i)T + (73.3 - 29.3i)T^{2} \)
83 \( 1 + (-0.182 + 0.752i)T + (-73.7 - 38.0i)T^{2} \)
89 \( 1 + (-0.395 - 2.75i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (-0.838 + 0.799i)T + (4.61 - 96.8i)T^{2} \)
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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

−11.32340673201835145621886734942, −9.901804307093124474405869326698, −9.305453613321362780990079911015, −7.63140495901763081683142872310, −7.23134827153188727848671913846, −6.22249160479574017399701648955, −4.92082767087679019646806126485, −3.86550989130346702430227054725, −2.46390303339597614348549534247, −0.975010137130991290882807093018, 2.87988011943281477411917351206, 3.36597394801982173973918532077, 4.96270569399854495647835947675, 5.63321756007822333475674513470, 6.54223315533996097365788402416, 8.070135007412567789675471742744, 8.722454286270307813322012715562, 9.794029609628561108384141026354, 10.75211007935080321363196263937, 11.44499503954537948071756696613

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