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Properties

Label 2-403-31.5-c1-0-11
Degree $2$
Conductor $403$
Sign $0.412 - 0.910i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
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  + 1.35·2-s + (0.0933 − 0.161i)3-s − 0.160·4-s + (1.08 + 1.88i)5-s + (0.126 − 0.219i)6-s + (−1.43 + 2.49i)7-s − 2.93·8-s + (1.48 + 2.56i)9-s + (1.47 + 2.55i)10-s + (1.47 + 2.55i)11-s + (−0.0149 + 0.0259i)12-s + (−0.5 − 0.866i)13-s + (−1.95 + 3.38i)14-s + 0.406·15-s − 3.65·16-s + (0.832 − 1.44i)17-s + ⋯
L(s)  = 1  + 0.959·2-s + (0.0539 − 0.0933i)3-s − 0.0802·4-s + (0.487 + 0.843i)5-s + (0.0516 − 0.0895i)6-s + (−0.543 + 0.942i)7-s − 1.03·8-s + (0.494 + 0.855i)9-s + (0.467 + 0.808i)10-s + (0.445 + 0.770i)11-s + (−0.00432 + 0.00749i)12-s + (−0.138 − 0.240i)13-s + (−0.521 + 0.903i)14-s + 0.105·15-s − 0.913·16-s + (0.201 − 0.349i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.412 - 0.910i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (222, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.412 - 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68372 + 1.08583i\)
\(L(\frac12)\) \(\approx\) \(1.68372 + 1.08583i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (4.88 - 2.66i)T \)
good2 \( 1 - 1.35T + 2T^{2} \)
3 \( 1 + (-0.0933 + 0.161i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.08 - 1.88i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.43 - 2.49i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.47 - 2.55i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.832 + 1.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.39 + 5.87i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.72T + 23T^{2} \)
29 \( 1 + 1.93T + 29T^{2} \)
37 \( 1 + (0.166 - 0.287i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.561 + 0.972i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.62 + 4.54i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.43T + 47T^{2} \)
53 \( 1 + (-5.94 - 10.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.23 + 5.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 5.91T + 61T^{2} \)
67 \( 1 + (5.73 + 9.93i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.218 + 0.377i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.33 - 4.04i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.84 + 11.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.91 + 6.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.45T + 89T^{2} \)
97 \( 1 + 1.01T + 97T^{2} \)
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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

−11.64627466464763148896221768157, −10.58433170996663962568295648907, −9.535364901505325667002273397389, −8.962719317101194259789700417232, −7.34534690654072487716668135194, −6.60050975300531248302495143426, −5.46611664515514538339214088747, −4.74697773939748819982877836944, −3.22946955578380940454790597860, −2.36829463064330519966885012285, 1.07438691195020510448966930793, 3.45200663145244014157335580653, 3.97657736903749057198568516328, 5.22564122735155324182882726269, 6.07195178456528597588380868478, 7.06958015599774504808832074381, 8.521923947031241840161898884865, 9.421258329306774342230708397162, 9.941614686609210642947399788072, 11.35094415136139511756288836579

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