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Properties

Label 1-344-344.323-r0-0-0
Degree $1$
Conductor $344$
Sign $0.996 + 0.0789i$
Analytic cond. $1.59752$
Root an. cond. $1.59752$
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  + (−0.623 + 0.781i)3-s + (−0.900 + 0.433i)5-s + 7-s + (−0.222 − 0.974i)9-s + (−0.222 − 0.974i)11-s + (0.900 − 0.433i)13-s + (0.222 − 0.974i)15-s + (−0.900 − 0.433i)17-s + (0.222 − 0.974i)19-s + (−0.623 + 0.781i)21-s + (0.222 + 0.974i)23-s + (0.623 − 0.781i)25-s + (0.900 + 0.433i)27-s + (0.623 + 0.781i)29-s + (−0.623 − 0.781i)31-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)3-s + (−0.900 + 0.433i)5-s + 7-s + (−0.222 − 0.974i)9-s + (−0.222 − 0.974i)11-s + (0.900 − 0.433i)13-s + (0.222 − 0.974i)15-s + (−0.900 − 0.433i)17-s + (0.222 − 0.974i)19-s + (−0.623 + 0.781i)21-s + (0.222 + 0.974i)23-s + (0.623 − 0.781i)25-s + (0.900 + 0.433i)27-s + (0.623 + 0.781i)29-s + (−0.623 − 0.781i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $0.996 + 0.0789i$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ 0.996 + 0.0789i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9323249844 + 0.03685530398i\)
\(L(\frac12)\) \(\approx\) \(0.9323249844 + 0.03685530398i\)
\(L(1)\) \(\approx\) \(0.8443826267 + 0.1127026356i\)
\(L(1)\) \(\approx\) \(0.8443826267 + 0.1127026356i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + (-0.900 - 0.433i)T \)
19 \( 1 + (0.222 - 0.974i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 + (-0.623 - 0.781i)T \)
37 \( 1 + T \)
41 \( 1 + (0.623 + 0.781i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (0.900 + 0.433i)T \)
59 \( 1 + (-0.900 - 0.433i)T \)
61 \( 1 + (0.623 - 0.781i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (0.900 - 0.433i)T \)
79 \( 1 - T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (-0.623 + 0.781i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.59697545321975888363395761788, −23.9745576290416613182627262359, −23.21501697196235671847968820938, −22.61898980723941208079408251902, −21.22179038867810314410848396246, −20.40854527965932405518065280286, −19.52366330322070297857091540471, −18.47378659092033147603472819650, −17.88439419919973499069762459125, −16.89681133470203071514728381781, −16.02747932414071593343566094745, −15.01069096467303619283933216231, −13.9589501709210247158097514943, −12.80799148927919323502090862341, −12.149448213156834368975982334555, −11.28136673234911787431581776889, −10.55659243983924903975341085682, −8.80404307939099525044966512424, −8.01813869103437430348988031726, −7.2201771639543860394694378517, −6.085908751519022370980334250310, −4.83870578458301855513350486887, −4.12795173787375597004327454025, −2.189983124555723860626619290493, −1.133958148402487885478331271264, 0.80203565023845672931646459064, 2.90275003073737152093480695424, 3.92971572066544585956693294965, 4.88348011728722960220524690793, 5.889640469545388057611459978553, 7.11102541574854975458248332888, 8.262781918541220157118420732275, 9.09706609571811047044583887276, 10.58605088156545215060700446515, 11.32499091107993965585682760866, 11.50523117684903646019031308925, 13.10978938323271853091926186643, 14.24851916511787873746253990558, 15.31415231870492298167787848903, 15.75283661419858331087680365890, 16.72279911152051101752992018846, 17.91785935909457270066610521851, 18.36605263202721331211274153908, 19.77584645918217058186824687058, 20.508214905845293094334626500294, 21.59889380223375894276184168649, 22.08838776920100787960165502994, 23.35537536240767123032666410219, 23.65015180578337017260965017727, 24.72053954791979364448482830900

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