Properties

Label 1-344-344.235-r0-0-0
Degree $1$
Conductor $344$
Sign $-0.993 - 0.116i$
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  + (0.733 − 0.680i)3-s + (−0.988 − 0.149i)5-s + (−0.5 + 0.866i)7-s + (0.0747 − 0.997i)9-s + (−0.900 + 0.433i)11-s + (−0.365 − 0.930i)13-s + (−0.826 + 0.563i)15-s + (−0.988 + 0.149i)17-s + (−0.0747 − 0.997i)19-s + (0.222 + 0.974i)21-s + (−0.826 − 0.563i)23-s + (0.955 + 0.294i)25-s + (−0.623 − 0.781i)27-s + (−0.733 − 0.680i)29-s + (−0.955 + 0.294i)31-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)3-s + (−0.988 − 0.149i)5-s + (−0.5 + 0.866i)7-s + (0.0747 − 0.997i)9-s + (−0.900 + 0.433i)11-s + (−0.365 − 0.930i)13-s + (−0.826 + 0.563i)15-s + (−0.988 + 0.149i)17-s + (−0.0747 − 0.997i)19-s + (0.222 + 0.974i)21-s + (−0.826 − 0.563i)23-s + (0.955 + 0.294i)25-s + (−0.623 − 0.781i)27-s + (−0.733 − 0.680i)29-s + (−0.955 + 0.294i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $-0.993 - 0.116i$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ -0.993 - 0.116i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02302574892 - 0.3946842669i\)
\(L(\frac12)\) \(\approx\) \(0.02302574892 - 0.3946842669i\)
\(L(1)\) \(\approx\) \(0.7183818783 - 0.2463761321i\)
\(L(1)\) \(\approx\) \(0.7183818783 - 0.2463761321i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( 1 + (0.733 - 0.680i)T \)
5 \( 1 + (-0.988 - 0.149i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.900 + 0.433i)T \)
13 \( 1 + (-0.365 - 0.930i)T \)
17 \( 1 + (-0.988 + 0.149i)T \)
19 \( 1 + (-0.0747 - 0.997i)T \)
23 \( 1 + (-0.826 - 0.563i)T \)
29 \( 1 + (-0.733 - 0.680i)T \)
31 \( 1 + (-0.955 + 0.294i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (-0.365 + 0.930i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (0.955 + 0.294i)T \)
67 \( 1 + (0.0747 + 0.997i)T \)
71 \( 1 + (0.826 - 0.563i)T \)
73 \( 1 + (-0.365 - 0.930i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.733 + 0.680i)T \)
89 \( 1 + (0.733 - 0.680i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−25.68409666807001978632078194739, −24.19428032987086480576519442668, −23.7040151206010122469165838564, −22.55109768314762152685629860545, −21.84683605967657657088204208767, −20.62958685136436091552489729465, −20.121341905829323453539777985377, −19.21752549921601742103542571288, −18.56494093695464725153817559077, −16.93147902929840453825013250152, −16.1485139543144950431303179892, −15.599374997230765331988390107105, −14.492725623690909872351107979577, −13.73032051566522229448452409892, −12.740118411877611879976273807503, −11.388847375637022076311916163739, −10.60821788115018408744881647665, −9.69905068940000581461065084207, −8.59064558344882502413721802234, −7.7027538462645586992411857751, −6.869550560955830216562585960016, −5.18775077190916361407571692400, −3.994675456518067580976815624697, −3.5268742267382709461569864021, −2.130740573291575533056100916892, 0.205223757254754594487492228, 2.225509605196901536522914152340, 2.97582743155521084330467374113, 4.23896920586618299454060440734, 5.57518618456871723678472689500, 6.8729685803689474415465287668, 7.728379610222435686822425877853, 8.54627461766455966813437395384, 9.40739492941499218016323788720, 10.75493833423304416665449443296, 11.97018787463360947503150844083, 12.72879698719907950760641682738, 13.27587488165788889234007124372, 14.80592240110534035192360279239, 15.36029777588921527495680825749, 16.03803336389888335077985133160, 17.656313293982273722624459286050, 18.34142437584350500731240343960, 19.24021820353590598904317509123, 19.95750554016691289123825544406, 20.57386094698895223312885014373, 21.918271286324780172780785800675, 22.79262576630340121735746265588, 23.793320910512906253986696573037, 24.42612831924637792493361646114

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